monte carlo simulation (wilmott varie)

8/6/2019 Monte Carlo Simulation (Wilmott Varie)

http://slidepdf.com/reader/full/monte-carlo-simulation-wilmott-varie 1/149



terest rate products by simulation:of risk-neutral paths

factorization:tion of Cholesky factorization for convertingd random variables into correlated random variables.

al integration:e Carlo integration

quence:of the Halton low-discrepancy sequences.



© Paul Wilmott 2001Pricing vanillas by simulation. Inputs in grey boxes

Asset 100 Time Sim 1 Sim 2 Sim 3 Sim 4Volatility 0.2 0 100 100 100 100

Int. rate 0.05 0.01 100.93 100.82 100.89 98.950.02 98.19 99.85 101.85 100.48

Strike 100 0.03 98.35 99.94 99.88 100.79Expiry 1 0.04 99.32 100.25 99.29 100.78

0.05 99.06 101.49 103.49 101.76CALL 9.6366 0.06 99.38 104.23 102.84 102.27

PUT 5.9350 0.07 97.13 107.43 104.77 101.310.08 92.87 105.37 107.15 101.71

Timestep 0.01 0.09 91.52 105.74 105.48 98.870.1 92.13 107.65 105.53 95.67

How big are the errors in the 0.11 90.8 109.48 106.7 94.72option values? 0.12 87.7 110.06 107.83 90.97

0.13 86.18 109.92 105.76 91.29Code this up in VB 0.14 87.94 110.81 104.84 93.59

0.15 86.41 112.13 102.79 93.680.94 117.27 118.42 105.47 120.230.95 117.68 120.68 106.2 117.750.96 117.68 123.62 110.86 122.730.97 125.31 125.13 109.5 120.080.98 123.69 130.18 107.47 125.320.99 125.93 128.82 108.42 128.86

1 125.35 127.94 108.81 127.72

CALL Payoff 25.35 27.94 8.81 27.72Mean 10.13

PV 9.64PUT Payoff 0 0 0 0

Mean 6.24PV 5.94



Sim 5 Sim 6 Sim 7 Sim 8 Sim 9 Sim 10 Sim 11 Sim 12 Sim 13 Sim 14100 100 100 100 100 100 100 100 100 100

99.01 100.25 99.96 97.01 101.1 101.97 101.86 99.08 99.2 100.2100.48 98.65 98.38 97.42 104.23 104.92 100.93 101.44 101.77 98.17101.67 99.4 97.94 97.4 102.16 105.46 100.63 100.44 98.13 96.92

99.86 100.38 97.62 97.48 102.48 105.29 102.4 104 96.39 98.24100.24 100.55 99.73 99.98 102.96 107.6 101.39 103.1 98.33 95.65

99.67 98.73 97.57 99.73 101.41 107.39 101.15 103.27 95.35 96.16100.3 100.5 99 98.17 102.68 107.54 99.14 103.15 98.32 95.1499.91 102.87 95.45 95.51 102.5 106.26 99.67 101.77 93.98 95.2998.82 102.25 98.11 96.13 99.15 106.47 103.28 102.32 93.98 95.09

100.08 103.25 99.12 99.61 100.93 110.02 102.87 100.89 96.44 92.45101.78 104.43 99.91 99.56 97.24 111.31 101.8 100.13 94.06 93.09

98.63 103.14 101.99 95.91 100.14 112.08 100.14 98.75 95.55 93.5198.85 108.15 103.07 96.88 98.76 113.11 100.61 96.6 94.21 94.996.38 106.13 100.92 97.94 96.47 110.07 103.57 96.68 95.03 95.3595.56 106.01 100.85 96.16 95.51 109.76 106.49 98.21 93.66 95.8997.09 107.63 96.83 99.09 124.84 112.72 106.43 124.54 101.75 120.07

97.2 107.86 94.36 99.92 124.98 112.43 109.29 123.16 100.81 122.4196.57 112.29 96.85 104.59 126.42 110.61 111.03 121.82 101.13 120.67

100.78 111.2 96.94 101.67 126.82 107.41 116.95 118.98 103.17 119.68101.34 113.7 97.26 101.18 124.62 104.82 116.81 116.32 102.95 121.88100.67 112.94 97.04 101.19 125.01 103.79 115.29 117.34 103.98 121.18

98.72 109.46 97.89 105.18 126.06 105.18 114.39 112.95 105.44 120.42

0 9.46 0 5.18 26.06 5.18 14.39 12.95 5.44 20.42

1.28 0 2.11 0 0 0 0 0 0 0




100.01 99.15 100.75 102.3 99 100.75 96.11 97.65 101.89 97.19101.55 97.74 99.61 104.19 95.89 101.71 98.81 97.36 101.12 95.94102.09 97.52 98.9 104.64 93.45 98.35 97.43 98.5 103.09 97.15

100.8 100.43 97.93 107.33 95.12 98.66 97.65 99.74 103.18 96.64102.42 102.04 101.17 105.68 92.47 102.9 99.39 100.9 102.5 97.45103.06 102 101.41 106.89 94.83 103.48 96.63 103.94 103 95.85104.59 102.19 101.99 107.6 93.45 100.69 95.84 101.18 102.29 94.13101.81 100.35 104.78 107.97 96.47 102.57 95.25 98.2 105.56 93.85

98.84 102.23 106.46 109.41 92.23 105.85 94.31 92.97 101.38 91.995.42 98.45 103.75 109.5 90.5 99.92 97.28 91.07 98.17 93.2996.08 96.21 105.26 110.05 88.85 99.85 99.48 88.29 93.94 95.46

100.42 95.93 108.01 107.66 93.65 98.26 100.99 88.88 92.74 94.397.82 95.02 111.16 108.17 91.79 97.59 103.68 89.05 94.35 93.88

95.7 94.01 111.05 108.23 89.72 100.12 107.12 88.91 93.37 95.6498.68 92.13 110.52 111.65 90.82 102.64 104.56 86.98 93.26 96.0393.57 100.51 99.27 127.92 106.1 137.42 125.22 107.6 96.39 93.4290.66 99.47 98.83 126.35 105.98 138.98 126.93 107.28 93.48 94.55

94 94.98 97.08 126.65 105.79 141.9 124.38 106.25 95.59 93.5798.48 92.69 99.15 131.46 105.68 146.65 124.04 105.64 97.17 94.9297.37 91.87 95.48 133.75 106.54 142.38 123.02 104.12 96.11 92.9597.09 90.26 95.45 138.18 104.11 142.51 122.18 105.03 97.91 97.498.32 88.2 90.72 136.23 102.35 142.49 121.42 107.58 96.55 96.32

0 0 0 36.23 2.35 42.49 21.42 7.58 0 0

1.68 11.8 9.28 0 0 0 0 0 3.45 3.68




99.34 100.93 99.62 98.64 100.51 101.38 96.61 100.89 97.47 103.92101.69 100.25 101.57 97.23 101.88 100.37 95.11 102.55 97.63 102.76106.26 100.66 102 97.11 101.72 98.73 94.52 100.49 99.22 100.26105.32 98.74 102.81 96.36 100.18 99.96 94.73 100.32 97.72 100.84108.24 98.37 104.78 97.36 104.19 98.62 96.03 97.45 97.55 100.36105.27 96.7 105.92 96.88 101.84 96.76 95 97.54 97.75 101.28106.02 94.59 104.68 94.19 100.28 98.85 97.31 99.93 96.96 99.46105.09 98.09 106.26 94.92 99 100.06 98.05 99.28 95.03 100.17108.84 98.96 106.5 92.89 99.18 102.74 97.93 103.25 94.67 99.99108.24 95.3 106.18 94.01 98.69 98.64 101.56 101.08 96.94 98.11113.76 95.32 106.01 93.16 97.25 98.59 101.09 97.92 98.22 97.86109.81 95.89 108.08 92.25 96.9 97.96 100.64 95.73 100.17 96.51109.72 99.09 106.74 92.45 94.92 93.83 102.16 96.34 98.88 96.77107.35 99.38 108.44 90.27 95.96 93.9 101.96 97.27 99.67 97.36110.77 100.92 109.59 92.58 99.12 91.77 101.25 98.46 101.33 96.65

98.26 99.38 121.36 85.17 80.03 86.19 87.17 103.93 79.91 92.4595.99 99.39 119.36 86.45 79.49 85.05 87.18 104.61 81.74 92.6396.43 98.83 117.92 85.97 78.51 82.31 87.24 104.14 82.19 91.3494.08 101.47 120.63 87.08 78.97 81 83.55 104.78 82.59 87.8495.61 101.37 117.93 87.57 77.59 77.88 83.88 104.58 82.91 86.5195.04 100.08 120.73 83.63 79.27 77.64 85 103.2 86.77 87.2593.77 101.64 123.35 84.46 77.12 78.73 83.84 104.42 85.97 88.69

0 1.64 23.35 0 0 0 0 4.42 0 0

6.23 0 0 15.54 22.88 21.27 16.16 0 14.03 11.31




98.46 101.11 99.82 99.66 101 98.67 97.5 104.5 100.65 102.3198.96 103.62 100.93 102.28 101.17 97.53 98.84 101.89 101.39 100.48

100.59 103.03 101.27 101.87 102.81 96.1 99.29 102.92 98.99 103.7198.92 103.93 101.71 103.75 104 96.03 98.35 101.33 100.02 105.17

103.93 105.08 101.61 104.54 107.41 93.34 97.79 98.8 99.57 108.47104.26 107.43 103.26 107.02 109.11 91.13 99.49 96.51 99.2 112.2103.28 106.75 101.21 108.75 110.31 89.59 98.42 95.84 100.07 113.74104.59 107.79 98.73 108.06 109.72 92.38 97.41 94.27 102.34 113.76105.55 107.06 101.84 105.45 110.88 91.82 96.82 92.33 101.51 111.57104.51 107.39 104.16 105 108.53 90.05 97.06 90.92 102.12 109.3106.01 105.73 103 105.94 106.33 88.45 99.28 90.44 99.73 111.28105.36 105.67 101.04 103.26 110.72 89.8 99.03 89.53 96.6 109.31104.06 105.53 102.27 103.13 105.82 88.39 98.61 87.66 97.95 113.15105.63 104.33 106.55 102.55 105.99 88.05 98.09 88.13 98.52 112.63107.42 107.47 108.08 101.37 105.9 85.57 96.68 87.8 100.18 108.13111.89 80.73 145.21 106.48 80.24 93.87 90.32 114.58 102.08 127.71108.23 81.19 143.82 106.06 81.15 94.76 90.93 109.32 104.84 123.63109.76 80.79 146.97 101.4 81.4 93.46 92.96 109.22 106.66 122.94112.84 81.92 154.52 99.89 77.77 91.76 93.76 110.07 108.43 116.21111.74 78.74 148.91 97.9 78.61 91.29 94.78 106.89 108.71 116.09114.34 77.92 150.59 98.27 77.03 89.97 96.28 104.8 110.44 122.36115.01 75.97 147.71 95.69 76.5 89.82 96.5 105.43 109.84 124.94

15.01 0 47.71 0 0 0 0 5.43 9.84 24.94

0 24.03 0 4.31 23.5 10.18 3.5 0 0 0




101.24 98.43 101.03 99.16 103.75 99.78 97.61 99.24 99.61 101.3998.71 95.4 96.86 100.53 103.06 93.66 98.05 101.69 101.73 101.28

99.8 96.9 95.67 98.29 102.94 90.63 98.8 99.56 102.77 102.88100.39 98.75 92.95 97.87 101.41 89.3 98.78 99.83 103.87 103.82104.38 96.97 92.53 99.25 101.52 88.28 96.03 97.85 103.59 104.62100.91 96.78 94.5 97.98 98.92 87.27 97.11 97.67 100.44 104.3100.97 97.66 92.81 99.82 95.43 88.81 100.35 95.85 101.28 106.96101.93 94.54 93.72 100.4 94.52 90.89 98.71 98.14 103.25 105.51

102.3 92.27 93.51 100.67 93.81 91.02 101.68 98.33 98.27 104.98104.04 93.97 91.73 101 96.53 88.96 101.55 97.37 99.67 109.41107.16 91.86 92.72 104.75 95.17 89.04 101.19 97.27 97.16 111.52103.65 93.4 88.65 102.88 92.6 87.28 98.99 96.07 94.9 118.59105.59 92.17 87.78 101.95 94.58 85.18 101.51 96.47 96.41 120.21105.36 90.33 89.94 104.81 93.95 82.89 104.08 94.94 96.42 121.87105.74 90.65 88.71 103.26 93.1 79.6 100.63 94.96 94.41 118.92132.13 86.51 94.9 102.23 99.51 75.53 115.6 108.51 102.74 101.21130.67 86.75 92.93 101.45 99.62 74.65 115.81 108.98 105.71 100.98128.74 89.78 89.91 103.43 97.58 75.18 113.69 106.37 105.28 101.85129.82 85.88 91 102.4 93.97 73.01 111.51 105.77 103.64 100.07128.33 84.75 91.38 105.23 96.7 71.26 113.89 107.14 105.1 100.82131.99 87.38 90.44 108.37 100.84 69.84 112.56 104.96 103.7 98.71130.66 85.84 91.75 112.15 106.19 69.67 110.23 107.43 99.33 99.29

30.66 0 0 12.15 6.19 0 10.23 7.43 0 0

0 14.16 8.25 0 0 30.33 0 0 0.67 0.71




98.92 99.05 100.24 99.85 99.27 100.38 98.61 99.12 95.35 101.2596.95 98.09 99.84 99.69 101.45 99.64 96.39 100.46 94.4 102.9995.29 97.53 100.95 97.34 99.97 96.43 93.37 102.12 93.14 101.6793.78 97.09 100.21 99.86 103.22 97.01 92.18 101.82 92.52 101.3296.13 97.54 97 99.33 103.18 94.94 94.22 97.14 90.24 100.7796.35 96.1 96.45 102.79 103.1 95.25 94.01 92.28 91.86 100.1696.52 95.48 95.45 101.46 102.99 95.84 95.67 89.62 91 98.3798.39 98.68 99.64 100.23 104.63 93.75 96.44 89.38 91.89 99.9298.71 98.41 100.74 101.76 104.87 93.12 96.23 90.39 92.75 96.3999.43 100.06 100.63 105.31 106.16 94.17 99 90.37 92.2 96.1697.61 97.44 100.65 104.23 105.79 92.15 98.31 92.17 92.07 94.14

98.4 98.1 101.06 101.26 105.04 91.94 98.2 95.9 91.65 96.6896.09 95.93 98.68 102.36 104.49 91.11 99.72 92.21 91.38 100.5798.74 94.82 104.42 99.79 102.95 89.71 101.38 94.29 88.03 98.3295.68 91.83 104.55 101.9 104.05 87.22 102.48 95.51 88.29 97.691.63 82.16 104.61 76.04 104.83 84.78 136.74 111.94 85.95 93.493.18 81.37 105.84 77.32 103.87 83.39 133.88 110.16 86.26 95.2193.68 81 103.11 77.29 100.62 79.65 134.47 113.41 86.01 94.41

93.3 81.22 104.68 76.54 99.66 79.95 137.1 114.44 84 93.1789.53 82.26 105.23 76.12 97.95 82.4 134.72 113.41 83.76 92.49

90.8 83.31 105.14 72.69 96.53 83.49 137.18 110.17 83.55 93.1688.99 82.53 104.99 70.95 99.95 82.41 137.33 111.26 87.36 92.78

0 0 4.99 0 0 0 37.33 11.26 0 0

11.01 17.47 0 29.05 0.05 17.59 0 0 12.64 7.22




101.2 95.27 99.1 98.26 100.05 98.79 99.36 99.48 104.4 95.1899.96 94.92 97.13 99.14 99.57 98.34 98.13 98.24 107.34 95.4297.62 95.83 98.4 95.6 100.66 98.6 99.61 95.11 105.76 95.2795.46 94.83 97.78 98.32 103.68 98.94 102.16 92.42 108.8 95.2995.68 95.48 99.26 95.66 105.27 97.87 104.58 94.85 109.64 92.4397.37 94.9 96.36 96.11 108.72 99.04 103.51 95.33 107.38 93.7296.31 94.58 95.33 94.31 109 98.22 104.51 94.27 105.07 91.0693.45 91.49 96.25 93.91 106.4 96.86 106.73 96.72 103.1 89.8193.56 90.69 93.48 91.83 108.64 95.85 104.36 97.5 101.22 88.3290.51 89.65 95.33 91.9 108.53 96.08 107.2 95.08 99.96 90.2588.97 88.28 95.7 92.51 103.85 97.17 109.59 91.52 101.22 88.0489.75 88.04 96.74 89.33 100.66 98 113.43 90.91 98.15 88.188.52 87.86 101.01 89.08 101.95 95.67 115.35 91.34 98.12 88.1493.09 89.59 97.01 89.97 101.17 96.81 111.36 93.51 95.24 88.2592.45 89.5 96.73 91.63 97.6 98.56 113.15 94.96 96.02 89.6

114.37 95.3 132.11 109.17 98.94 122.02 115.34 97.36 117.34 97.03115.26 95.13 130.84 105.03 98.46 122.34 114.29 95.94 118.99 100.93115.82 92.58 126.53 105.67 97.76 123.1 113.14 98.04 117.47 99.87113.66 92.08 120.97 105.74 101.77 119.67 115 101.26 119.59 99.11117.03 91.18 120.57 102.11 99.6 124.66 117.7 100.82 118.14 96.99

119.9 89.98 120.73 102.17 99.82 118.09 125.25 104.47 115.71 99.64120.49 89.35 119.67 105.44 102.5 121.29 124.12 110.97 116.42 101.48

20.49 0 19.67 5.44 2.5 21.29 24.12 10.97 16.42 1.48

0 10.65 0 0 0 0 0 0 0 0




99.93 102.05 96.56 100.34 99.44 99.1 100.78 97.34 99.75 103.21101.18 103.27 98.26 102.78 100.08 98.44 97.93 95.66 98.91 101.81

98.3 104.11 96.73 100.97 102.72 99.54 99.94 94.72 101.31 100.7793.62 105.77 95.47 100.43 104.55 101.79 98.4 92.22 100.39 101.1295.92 103.8 94.12 99.42 101.54 101.01 101.01 91.65 99.49 100.7995.11 103.6 94.61 98.71 100.25 102.07 102.71 89.6 100.89 102.3198.79 101.83 93.97 97.87 103.16 101.75 100.68 91.12 100.29 102.2499.21 102.46 92.1 98.85 103.02 101.58 101.36 91.31 100.69 10299.51 102.25 90.81 102.96 101.87 100.46 102.82 94.46 98.24 102.38

101.73 101.23 95.25 99.67 98.65 101.9 103.75 96.82 96.46 101.52101.01 104.05 94.42 103.46 101.85 101.84 103.81 97.43 97.13 102.03

98.49 104.93 92.86 104.34 101.07 103.12 107.99 97.58 97.98 105.31100.08 104.89 92.63 108.55 98.1 102.87 107.67 102.63 96.92 101.43

99.69 103.51 94.77 111.23 98.57 99.86 106.91 103.01 95.55 101.9999.44 102.15 94.33 112.34 100.17 102.86 102.41 102.54 96.04 104.3385.99 136.88 105.84 94.85 168.41 101.31 82.57 122.85 110.74 120.74

85.8 134.05 104.37 92.09 172.25 99.84 83.65 119.19 110.52 122.0786.48 139.99 107.58 91.63 173.5 101.23 84.86 117.72 109.28 123.7788.71 140.96 108.71 92.01 169.75 100.74 84.08 116.99 108.9 120.5290.27 143.52 111.33 90.86 174.42 99.95 82.63 113.18 107.54 119.07

90.4 146.18 108.26 91.92 175.9 97.4 82.35 112.85 109.63 114.4588.52 142.98 108.84 93.43 180.41 101.9 82.59 111.92 107.36 117.21

0 42.98 8.84 0 80.41 1.9 0 11.92 7.36 17.21

11.48 0 0 6.57 0 0 17.41 0 0 0




102.82 97.9 101.2 101.37 101.75 99.83 102.4 101.33 101.86 99.04101.71 95.05 101.37 98.66 102.03 101.72 100.29 102.72 104.97 100.69103.76 93.38 102.67 99.7 104.08 103.06 103.41 102.59 106.8 103.9101.52 92.35 102.08 98.2 104.21 101.19 104.01 103.04 108.97 101.92101.96 94.01 102.4 102.44 105.19 100.15 102.81 105.18 111.55 106.43104.18 95.52 105.88 104.3 103.63 99.56 99.32 107.02 110.7 105.6104.63 99.44 107.98 104.83 99 96.84 100.33 107.56 113.75 99.71106.54 100.77 106.48 104.05 100.67 94.86 102.14 106.02 111.49 97.96110.28 101.47 104.77 110.33 98.17 94.56 101.74 106.01 107.37 98.67106.74 101.14 102.89 111.22 98.5 94.18 103.05 109.38 107.64 96.18108.62 100.93 104.74 115.02 95.73 92.38 101.49 107.53 108.69 94.1107.31 102.12 105.23 117.05 93.65 91.87 100.86 105.22 112.12 94.25105.03 100.7 106.92 114.04 94.14 91.25 101.02 104.28 113.45 91.81108.99 99.78 103.7 117.57 92.4 91.97 101.13 103.15 110.72 93.54108.81 97.48 103.7 122.54 91.03 90.01 100.41 101.9 107.44 90.32121.05 77.69 128.7 98.63 121.71 98.12 98.48 83.65 96.71 108.6118.91 77.36 130.54 97.21 124.1 96.55 98.64 84.61 94.12 107.23121.64 74.08 134.11 96.98 123.11 96.66 99.65 84.2 96.75 106.1121.11 73.74 135.06 95.97 123.61 98.42 98.15 83.32 93.04 105.37121.67 74.25 132.98 98.1 122.67 99.48 102.68 83.11 92.33 100.47120.66 76.38 133.69 96.82 119.54 97.01 102.99 83.01 88.06 104.6122.42 75.24 139.24 96.42 122.7 100.03 99.86 84.86 87.51 104.02

22.42 0 39.24 0 22.7 0.03 0 0 0 4.02

0 24.76 0 3.58 0 0 0.14 15.14 12.49 0




98.49 98.01 97.93 99.06 98.7 98.66 100.94 97.64 100.33 100.86100.05 96.06 98.9 95.4 98.26 98.36 98.33 95.99 100.23 102.78

99.68 97.7 97.64 95.69 99.22 98.73 97.14 95.55 101.68 106.8399.87 101.66 96.99 94.69 97.12 97.35 96.33 96.28 99.79 105.75

101.38 100.15 96.66 94.22 100.21 98.04 96.6 95.62 100.5 104.6102.04 99.7 96.69 93.45 101.17 99.42 93 93.81 98.22 105.73100.23 101.74 95.94 99.18 102.49 97.35 93.89 91.61 95.83 107.56100.33 101.23 97.09 99.11 101.36 96.62 94.51 92.16 98.91 106.78

100.9 104.08 96.32 101.17 100.22 95.15 92.89 94.27 97.63 105.32101.27 105.48 94.56 101.98 99.3 92.38 94.23 92.28 97.81 106.99100.58 107.81 95.07 101.38 98.84 90.73 92.24 88.28 100.44 107.31100.45 107.54 93.52 103.71 97.58 90.22 93.92 91.41 102.13 107.02

99.51 104.58 91.98 99.74 94.68 87.31 93.08 92.48 103.07 105.78100.1 106.38 95.08 100.24 92.08 87.74 90.71 92.82 105.16 104.1799.59 107.38 96.61 100.58 90.48 88.55 87.39 92.25 104.74 106.83

108.19 110.71 110.3 122.82 85.91 56.83 107.49 93.56 157.31 82.24109.1 112.77 114.33 126.07 83 56.56 105.49 96.55 157.33 82.14

112.02 115.47 114.81 124.11 81.81 56.79 108.82 96.8 156.19 80.52111.33 116.15 115.77 122.29 83.34 58.9 107.46 92.04 158.61 83.54112.51 118.24 113.07 122.3 82.03 59.02 102.07 96.57 160.19 84.3114.63 117.18 111.94 124.9 81.71 59.3 104.4 93.73 158.86 83.92117.15 118.11 111.28 123.23 81.99 57.29 101.88 94.03 159.06 82.75

17.15 18.11 11.28 23.23 0 0 1.88 0 59.06 0

0 0 0 0 18.01 42.71 0 5.97 0 17.25




97.1 97.28 98.97 99.1 100.28 98.9 100.51 96.53 99.72 97.6792.72 102.84 99.76 98.2 100.83 101.3 100.37 96.62 100.44 96.4592.15 98.38 96.19 97.08 99.83 101.84 98.19 94.01 98.9 93.4293.25 98.2 98.13 99.1 100.22 99.65 96.41 93.67 97.62 90.3191.87 99.13 98.05 100.4 100.01 99.71 95.76 94.33 95.71 91.3889.63 98.5 97.63 96.19 98.12 100.13 94.54 94.3 95.36 91.2991.39 102.02 96.54 98.58 97.38 97.99 94.61 95.96 94.33 93.6290.01 101.2 98.02 94.56 95.35 100.69 95.61 95.93 93.83 94.2694.04 103.64 98.73 95.72 96.89 100.07 96.86 95.8 92.86 92.7593.78 103.67 101.04 92.18 96.86 101.01 95.24 96.15 92.94 92.698.12 104.46 104.06 93.8 96.93 102.04 93.67 94.11 92.45 94.68

101.88 105.9 104.75 92.39 98.17 102.75 91.89 94.37 94.53 92.85102.14 106.05 105.13 93.71 97.75 102.94 91.44 94.35 95.66 93.07101.92 104.67 102.63 96.46 97.31 104.8 93.83 94.82 98.67 96.48103.73 103.01 100.48 94.33 99.61 107.22 91.91 95.19 96.15 96.12121.04 100.51 102.6 108.14 112.11 132.55 112.3 120.73 107.69 95.31115.18 99.7 103.17 107.4 110 131.94 110.94 124.68 111.09 93.23111.33 101.4 100.44 105.5 109.61 131.06 110.48 124.54 115.65 90.66117.62 101.71 97.13 105.6 110.11 129.07 112.8 122.64 113.74 89.24114.41 100.93 99.07 103.64 107.8 124.37 117.16 128.62 112.08 89.09112.91 102.45 97.66 103.01 105.51 124.23 116.58 126.13 107.88 91.73111.98 102.06 98.94 104.56 103.82 126.87 113.9 123.78 107.36 91.09

11.98 2.06 0 4.56 3.82 26.87 13.9 23.78 7.36 0

0 0 1.06 0 0 0 0 0 0 8.91




100.91 101.8 102.12 99.66 98.23 99.12 100.56 102.1 100.08 96.8103.5 104.52 103.23 101.69 99.91 100.49 100.23 104.75 100.84 94.13

105.86 103.82 101.89 102.1 99.69 102.86 98.17 102.91 97.41 93.28106.92 104.02 104.12 102.4 97.46 99.6 98.76 102.47 98.35 94.13103.86 104.56 104.3 100.24 99.2 97.85 97.71 105.02 97.1 94.55103.24 109.19 105.84 99.57 101.33 96.38 99.75 107.54 95.54 94.61100.04 110.66 105.3 102.1 104.81 92.84 101.13 105.33 94.73 92.93102.05 110.26 104.41 99.01 102.16 94.48 99.47 103.41 96.52 91.86101.49 107.66 104.76 96.34 102.77 93.33 101.72 102.65 101.39 93.79

99.54 112.65 106.28 95.38 101.34 95.29 101.32 103.18 102.06 92.695.97 114.74 104.77 91.92 100.14 98.55 106.26 102.07 103.11 93.92

95.3 115.2 103.15 93.26 98.7 98.06 108.7 106.24 100.85 94.597.8 116.25 99.85 90.47 102.03 100.76 106.34 107.22 101.87 98.88

96.34 116.53 103.24 93.92 103.42 97.6 108.4 110.22 101.2 100.4695.64 114.04 104.99 91.76 101.44 98.31 110.47 109.18 100.98 100.67

130.81 98.54 101.41 78.03 93.12 107.28 103.3 130.8 107.55 112.1128.24 98.9 103.78 78.32 92.65 108.22 101.36 130.65 108.4 111.63128.54 97.1 103.44 81.54 92.67 108.87 100.52 131.49 107.15 113.01125.38 97.6 106.19 82.52 89.16 109.42 100.44 127.97 109.65 116.12125.04 96.26 105 80.3 89.57 109.84 102.45 126.77 109.66 114.46128.93 94.86 108.16 79.37 89.77 114.46 98.83 126.47 107.58 113.13127.88 96.04 108.64 79.32 88.09 111.03 97.17 128.66 107.15 114.36

27.88 0 8.64 0 0 11.03 0 28.66 7.15 14.36

0 3.96 0 20.68 11.91 0 2.83 0 0 0




99.79 99.55 98.54 97.88 97.69 101.83 103.1 99.53 101.21 100.8999.72 97.16 96.18 97.97 99.37 102.58 98.43 100.75 105.16 97.4199.26 97.19 98.2 98.42 96.19 106.19 101.03 100.55 105.06 97.44

101.68 95.37 97.48 98.14 97.99 104.5 101.16 98.08 105.12 97.2898.32 97.52 96.3 96.58 96.92 105.26 108.39 98.43 106.48 98.2395.37 96.51 98.39 95.32 98.86 106.09 105.91 95.07 107.44 99.892.99 96.5 98.88 97.49 100.75 109.47 107.91 96.1 107.94 99.7693.02 96.78 98.51 97.69 98.91 109.88 106.24 96.76 107.34 97.0792.78 96.96 98.88 98.25 96.98 109.38 105.49 97.6 106.01 100.1196.14 96.46 96.72 97.34 97.05 108.73 103.75 94.17 105.59 100.2792.04 92.67 94.57 100 99.54 111.24 104.65 92.43 107.1 100.76

95.2 93.28 92.27 101.34 100.58 111.64 106.21 92.16 106.28 99.4398.11 93.32 94.6 98.64 101.92 109 104.78 90.87 107.43 97.0197.39 94.56 98.14 98.21 100.66 107.66 101.67 88.15 108.43 97.5498.09 92.95 96.71 97.5 101.09 105.96 105.02 86.14 108.64 98.1390.39 142.14 108.27 81.75 91.31 82.71 89.54 105.44 89.02 93.9692.19 139.22 107.97 82.15 91.64 83.03 93.64 106.86 87.33 94.7592.65 139.85 110.6 84.36 90.84 83.72 95.58 109.45 89.74 95.8592.83 138.91 107.5 85.28 88.87 83.35 95.15 112.04 88.8 95.7591.52 139.21 107.14 87.71 87.31 82.37 94.1 109.57 87.17 96.2894.74 138.39 105.67 86.86 89.28 84.86 93.66 108.12 86.7 93.9193.79 141.09 108.89 87.63 90.84 83.91 91.04 109.48 87.22 96.85

0 41.09 8.89 0 0 0 0 9.48 0 0

6.21 0 0 12.37 9.16 16.09 8.96 0 12.78 3.15




100.71 98.42 101.32 99.49 102.75 104.12 101.09 101.41 101.81 100.24102.13 99.34 100.33 98.99 103.04 104.18 100.07 99.54 101.05 100.8100.26 98.63 99.32 98.6 102.76 103.96 98.67 99.85 106.42 99.68100.84 99.57 101.11 96.75 101.27 104.49 101.22 102.16 106.97 100.77104.09 97.56 97.85 95.27 99.95 107.95 101.7 105.37 106.07 103.3101.85 97.85 97.58 94.78 102.32 105.98 99.14 101.77 110.46 103.85

99.38 99.51 95.3 96.88 103.27 107.84 97.67 103.85 113.45 102.998.37 100.86 99.61 97.74 102.68 108.96 98.74 102.76 112.67 102.9499.43 101.14 102.32 97.2 101.43 108.46 95.23 101.5 107.75 98.95

100.85 99.79 97.79 97.91 99.53 105.64 94.21 98.07 106.25 101.4297.94 100.2 101.32 98.35 102.74 109.24 97.3 97.15 103.76 103.696.22 97.72 103.23 97.93 102.59 107.86 101.24 98.01 103.73 99.03

96.4 100.86 104.81 98.94 98.59 107.23 102.64 98.94 101.77 97.3197.65 101.97 108.59 97.58 99.09 110.16 103.61 100.05 105.59 96.57

97.6 102.03 109.97 97.13 96.45 111.94 103.01 100.33 103.21 98.2486.15 79.09 75.13 64.29 94.09 105.38 90 117.74 107.31 83.84

87.4 81.04 73.66 63.05 87.48 107.12 90.98 121.32 106.36 84.2488.25 79.92 75.46 65.75 85.63 109.74 88.31 121.75 106.99 84.4684.81 77.81 76.38 66.93 86.52 106.77 87.67 123.83 106.61 83.24

84.9 80.26 73.05 67.76 85.82 108.74 88.33 122.47 107.36 83.6682.84 81.18 72.88 65.78 85.38 112.48 88.55 123.75 106.13 84.984.51 79.23 71.15 65.89 86.08 111.29 90.88 123.62 104.29 84.95

0 0 0 0 0 11.29 0 23.62 4.29 0

15.49 20.77 28.85 34.11 13.92 0 9.12 0 0 15.05




99.02 101.65 97.46 100.17 99 100.83 99.73 97.17 99.63 95.69101.27 100.25 101.38 101.48 100.11 99.78 103.94 97.91 98.92 95.25102.15 99.79 99.89 101.09 99.67 98.36 103.73 99.36 96.8 96.17100.89 101.27 98.44 104.34 98.18 98.32 107.53 98.81 98.35 95.95

99.75 100.04 97.03 100.02 100.86 97.6 111.13 99.81 97.09 95.58101.62 100.25 96.57 99.98 103.57 99.49 111.81 98.84 94.37 97.38102.34 101.4 94.54 96.53 99.73 100.48 112.86 99.06 94.7 97.17102.67 99.54 95.94 97.27 100.49 98.93 113.95 100.39 95.55 94.08102.02 100.7 96.09 94.9 100.39 99.55 112.47 105.14 97.38 93.72

100.1 103.31 95.64 96.95 98.04 98.97 106.69 106.36 98.7 94.5298.12 102.45 94.55 100.19 98.11 100.03 105.48 104.86 98.72 93.43

97.4 105.33 99.64 105.94 96.25 102.55 103.33 105.31 100.53 92.72100.06 107.64 99.32 104.45 96.07 102.61 104.85 104.95 98.27 91.53

100.7 106.57 96.51 106.55 94.29 102.55 107.78 105.05 102.64 89.81101.12 105.67 94.77 102.97 91.75 103.19 105.96 103.89 105.33 90.63

87.62 105.31 116.28 110.61 68.2 116.28 111.58 116.53 86.5 91.5187.84 106.03 115.08 111.62 67.87 114.21 114.44 116.99 84.2 88.1591.02 102.09 115.71 110.34 70.2 115.52 112.7 115.33 84.53 86.1794.27 101.94 116.19 108.15 67.51 112.13 110.75 115.26 83.1 87.894.07 101.67 117.44 105.37 66.97 111.15 112.93 116.17 84.36 90.3592.46 103.46 118.29 102.93 68.67 107.05 110.25 118.13 82.59 93.1590.38 105.1 122.88 105.36 69.62 108.41 111 121.23 83.82 91.95

0 5.1 22.88 5.36 0 8.41 11 21.23 0 0

9.62 0 0 0 30.38 0 0 0 16.18 8.05




100.48 97.58 101.61 97.43 106.56 99.22 101.74 97.2 96.83 98.0299.55 95.83 100.91 95.26 105.22 95.5 103.2 97.85 97.34 98.48

103.14 94.45 103.93 96.23 105.23 96.01 105.13 98.87 97.34 94.93101.43 94.96 103.83 97.54 101.72 97.88 105.68 99.38 96.07 96.46100.99 95.37 108.05 93.81 99.3 93.5 105.76 98.79 99.63 96.36

99.61 92.78 110.1 91 99.09 90.34 109.06 98.32 97.01 93.41102.4 90.99 108.57 91.29 101.25 91.14 109.72 96.53 95.49 93.65

102 93.75 108.51 90.93 102.75 91.9 109.49 99.85 94.33 91.65100.68 91.2 109.61 89.9 99.75 91.21 108.8 99.38 91.56 94.42

99.94 92.52 110.03 85.9 98.79 92.27 110.47 95.08 91.32 92.8299.77 94.95 109.14 83.39 95.83 92.39 106.5 94.56 89.58 94.6499.16 93.69 109.61 86.15 93.37 92.66 104.26 91.08 88.68 94.5497.59 92.81 109.58 88.54 93.63 94.67 104 91.38 87.68 96.0298.94 93.64 109.83 89.15 96.05 95.47 104.07 94.8 89.96 93.97

98.1 91.68 108.26 87.62 99.65 97.61 106.24 96.48 86.87 95.53107.96 65.55 101.42 77.04 92.26 104.69 105.52 100.96 86.34 81.18110.52 65 101.64 75.95 90.42 104.41 105.83 104.72 85.65 82.27111.24 67.83 104.95 75.56 89.22 108.36 110.28 105.66 83.72 80.76110.11 65.95 109.61 74.29 91.16 105.85 111.05 105.95 84.45 83.45111.96 67.19 104.16 73.73 88.96 109.36 112.55 107.14 85.68 83.13110.66 67 102.88 75.47 89.29 111.17 112.04 107.06 86.9 85.32108.79 66.97 109.9 75.88 89.66 109.73 111.46 108.85 87.67 87.56

8.79 0 9.9 0 0 9.73 11.46 8.85 0 0

0 33.03 0 24.12 10.34 0 0 0 12.33 12.44




101.12 100.9 100.92 98.98 103.43 101.6 96.77 100.72 98.41 98.1598.86 100.34 102.77 100.06 104.23 106.51 94.75 99.76 95.43 101.14

100.73 101.72 104.41 97.65 104.77 102.68 97.74 98.78 94.37 97.94102.08 100.06 107.68 97.87 108.88 103.14 94.77 98.57 92.53 96.03100.14 100.4 108.4 97.37 113.54 104.71 93.83 95.31 91.47 99.81101.43 102.63 108.4 96.23 111.82 106.21 93.86 95.78 93.29 99.98

98.48 99.37 110.2 95.79 106.56 105.22 92.44 97.25 94.68 100.03101.63 99.52 108.87 98.1 105.52 102.35 91.26 98.33 94.41 99.94

102.3 101.15 110.85 97.63 106.63 101.94 90.4 95.93 95.84 102.24103.57 100.99 109.34 96.74 106.87 102.64 90.38 98.58 94.44 101.26101.54 101.7 105.31 97.06 110.9 103.4 91.42 98.39 93.66 101.79

101.8 98.54 106.84 98.04 108.09 99.11 91.82 99.5 95.68 102.78105.37 97.38 107.16 96.55 111.15 99.48 91.8 103.35 98.69 102.81108.61 98.96 107.71 95.23 110.55 96.67 88.99 107.02 97.24 105.92109.07 96.28 110.84 95.85 110.89 99.49 91.16 106.07 94.24 107.01100.91 111.77 88.66 88.34 125.61 100.22 64.54 113.78 94.87 169.53

99.85 112.6 89.67 88.83 126.64 99.11 63.84 116.02 93.38 173.01103.17 112.49 93.31 90.33 130.44 103.76 64.93 113.58 92.07 172.16104.68 110.29 94.53 87.71 131.5 107.38 64.58 116.56 92.57 167.39107.26 107.56 93.66 86.19 132.04 106.7 64.97 115.66 93.81 166.66110.29 108.03 93.29 86.31 135.14 107.49 64.71 112.38 90.38 164.7110.15 107.8 91.79 84.77 138.14 101.91 63.69 114.13 87.87 165.48

10.15 7.8 0 0 38.14 1.91 0 14.13 0 65.48

0 0 8.21 15.23 0 0 36.31 0 12.13 0




103.39 100.83 101.51 97.78 103.26 100.7 98.26 100.16 100.83 99.85101.13 98.41 103.34 98.32 102.89 102.1 96.21 102.03 98.89 98.18100.16 100.33 106.93 99.58 99.7 103.14 97.3 105.07 96.53 98.71

96.25 97.56 107.32 97.87 96.81 106.89 96.96 105.24 97.81 99.8899.5 99.47 107.68 94.52 96.99 105.56 93.42 105 98.81 97.13

98.43 96.8 106.18 98.61 96.53 105.91 96.76 105.64 97.97 96.4897.23 97.16 106.35 100.25 94.16 106.17 97.47 103.82 99.7 97.3398.16 94 103.61 97.96 97.87 106.24 95.96 102.05 98.36 99.697.46 93.63 103.17 99.04 95.24 107.87 95.7 97.71 96.78 105.1496.88 94.52 101.48 98.27 97.53 110.15 98.65 97.04 97.28 105.9397.72 91.91 104.07 97.66 99.61 107.66 99.04 94.36 98.32 102.2195.26 94.32 103.56 96.62 102.83 106.77 98.75 95.55 98.14 101.7494.79 92.94 102.59 93.59 101.11 104.16 98.58 97.01 100.62 102.7991.91 94.72 103.16 93.45 102.18 103.53 99.44 99.39 104.12 104.0592.61 96.4 99.76 90.71 100.63 104.84 100.52 97.31 105.47 99.3278.55 114.04 96.36 124.07 116.19 108.99 95.45 88.27 129.25 82.6979.18 115.16 96.46 123.98 119.17 108.25 93.22 91.64 126.39 82.8878.84 118.49 93.75 124.67 116.9 113.88 94.32 91.2 124.98 80.881.68 116.41 92.69 126.79 114.99 115.62 96.36 90.11 123.1 83.683.39 113.46 92.03 125.67 114.85 117.14 96.16 88.79 119.92 85.3485.38 114.92 90.23 126.89 116.51 117.27 94.84 89.36 119.06 85.67

88.3 116.57 87.64 126.05 115.36 114.33 96.77 88.02 120.36 84.44

0 16.57 0 26.05 15.36 14.33 0 0 20.36 0

11.7 0 12.36 0 0 0 3.23 11.98 0 15.56



Sim 245 Sim 246 Sim 247 Sim 248 Sim 249 Sim 250 Sim 251100 100 100 100 100 100 100

100.88 101.56 99.1 101.76 99.24 98.62 97.7398.59 103.43 98.86 105.87 98.47 95.05 96.6999.01 102.82 99.62 104.41 103.28 93.01 95.72

101.69 103.74 99.59 104.66 106.25 92.12 97.298.63 104.23 97.72 106.57 107.05 92.76 97.6795.34 105.01 99.41 107.99 105.89 91.06 100.6796.96 112.31 97.99 108.23 102.66 91.95 100.297.05 110.88 97.23 107.89 102.37 94.79 97.1495.62 111.2 98.05 104.55 100.95 95.71 97.83

94.8 108.16 99.76 103 100.53 93.56 98.3198.38 108.17 95.4 104.27 98.37 96.29 96.7598.25 107.42 96.41 102.67 98.34 98.9 96.59

102.82 110.04 97.48 104.59 96.94 95.28 95.89103.62 115.19 100.34 105.77 94.34 97.24 98.12102.99 117.84 98.87 106.34 94.65 97.34 96.53

69.55 140.03 98.02 97.79 75.25 78.79 126.7869.61 138.84 101.42 95.46 75.65 75.7 128.2670.17 140.77 101.54 99.04 75.11 76.73 128.5270.25 135.52 101.56 100.62 76.15 77.19 125.6971.36 134.27 100.26 104.78 76.76 78.28 127.1471.02 132.02 101.28 105.45 76.7 77.67 132.3868.79 130.07 100.31 108 76.27 76.78 132.81

0 30.07 0.31 8 0 0 32.81

31.21 0 0 0 23.73 23.22 0



© Paul Wilmott 2001Pricing exotics by simulation. Inputs in grey boxes

Asset 100 Time Sim 1 Sim 2 Sim 3 Sim 4Volatility 20% 0 100.00 100.00 100.00 100.00

Int. rate 5% 0.01 95.63 100.18 101.16 100.090.02 93.34 100.80 101.62 99.37

Strike 105 0.03 94.58 100.06 102.36 94.10Expiry 1 0.04 96.40 102.11 97.52 90.12

0.05 97.46 104.99 99.91 91.05Exotic? 3.8341 0.06 94.07 103.41 100.23 90.33

0.07 91.86 103.85 99.97 91.48Timestep 0.01 0.08 91.12 103.45 99.56 89.54

0.09 91.82 103.94 97.69 89.46What type of option is 0.1 92.24 103.21 98.70 87.31this valuing? 0.11 93.05 102.90 102.65 86.30

0.91 88.69 80.78 93.51 90.02

0.92 87.73 81.72 93.59 86.910.93 89.94 82.27 95.79 87.330.94 89.78 84.38 94.17 85.720.95 91.05 88.50 93.98 85.320.96 90.59 88.61 97.42 84.720.97 91.50 84.85 98.14 83.900.98 91.00 84.95 97.24 84.690.99 88.62 84.13 94.39 81.66

1 90.08 84.05 95.29 83.30

Average asset price over each path 86.46 93.23 101.85 90.04

Payoff 0.00 0.00 0.00 0.00Mean 4.03

PV 3.83



Sim 5 Sim 6 Sim 7 Sim 8 Sim 9 Sim 10 Sim 11 Sim 12 Sim 13100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00105.08 102.77 101.02 101.14 102.46 100.54 100.43 100.08 98.95102.45 105.29 103.21 100.94 101.10 101.68 96.69 102.31 101.76104.72 105.33 100.16 102.16 103.39 98.46 97.24 106.28 99.56105.47 104.05 100.49 99.77 106.10 96.23 97.21 107.43 98.99107.89 103.10 101.24 100.41 101.92 94.72 93.70 104.66 97.21109.34 101.64 95.80 98.27 103.58 96.08 90.77 101.79 96.91110.53 99.11 95.25 98.83 105.20 96.54 93.60 98.84 100.99111.25 101.79 93.84 101.35 111.08 95.56 91.04 101.12 101.44113.97 99.44 95.28 103.24 111.33 95.38 93.28 103.47 99.95113.69 97.82 95.12 104.66 113.94 91.48 98.32 105.08 97.07110.49 100.43 95.67 102.47 113.51 92.57 97.71 105.32 92.41116.46 185.24 138.41 120.61 108.26 81.39 94.94 93.89 97.94

118.14 184.57 135.34 121.91 107.37 81.42 96.11 95.61 101.60117.68 185.54 132.79 122.92 106.67 82.08 93.22 92.94 98.04117.08 186.71 133.72 123.66 106.43 83.83 92.84 93.95 98.04111.25 179.63 134.67 125.81 106.01 81.67 94.12 95.05 96.68111.83 183.29 128.75 123.93 104.63 82.81 97.28 96.10 95.30109.68 184.24 127.92 124.25 104.18 83.02 97.12 96.91 95.80112.00 187.59 129.71 124.70 102.26 79.70 96.22 99.67 96.16109.49 192.22 132.47 126.96 102.39 79.28 96.68 100.70 93.54105.96 183.66 130.68 126.23 103.23 79.13 95.78 103.10 96.05

117.72 138.71 123.53 107.10 109.55 86.04 96.40 101.73 98.20

12.72 33.71 18.53 2.10 4.55 0.00 0.00 0.00 0.00



Sim 14 Sim 15 Sim 16 Sim 17 Sim 18 Sim 19 Sim 20 Sim 21 Sim 22100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00100.24 99.09 100.09 97.72 95.22 98.76 103.43 103.49 98.85102.02 102.21 99.73 98.57 93.99 96.26 103.14 104.12 99.99102.00 99.82 99.48 96.70 93.36 95.16 100.38 109.37 101.33100.03 94.92 99.72 98.20 93.29 99.70 102.34 108.78 103.88101.78 94.73 100.21 97.71 92.43 95.75 103.76 108.59 103.67100.10 96.02 102.04 95.48 91.67 94.08 105.34 106.77 103.72100.80 94.46 101.35 93.12 92.72 93.15 109.39 103.68 104.06

98.56 92.39 95.89 91.58 93.21 90.39 112.42 101.77 105.3497.41 90.21 93.19 92.27 92.00 88.83 109.41 99.16 108.69

100.41 90.66 91.69 91.66 92.02 88.29 108.88 95.80 110.14100.65 89.47 92.51 92.85 95.65 88.47 106.16 97.91 111.43119.37 84.10 112.55 103.86 101.43 94.71 151.11 142.43 124.49

118.43 83.23 113.70 105.36 99.72 92.63 154.55 141.57 125.92118.35 84.88 120.05 105.98 96.27 91.62 151.71 142.58 126.48119.05 87.95 120.78 103.32 96.40 89.44 157.06 142.03 127.96117.97 85.53 124.14 103.94 94.45 89.03 158.77 142.32 129.77117.43 86.13 123.77 103.37 94.94 90.83 158.08 140.62 130.21114.49 83.04 124.46 103.32 97.54 89.86 157.08 140.44 128.56115.12 81.03 120.60 102.25 98.26 94.09 153.03 140.09 126.82114.95 80.87 119.36 99.03 96.64 94.07 150.44 138.37 129.37119.02 85.36 115.58 96.73 97.49 94.06 148.46 141.95 124.54

100.89 80.84 104.81 95.39 96.13 89.16 122.15 115.60 113.95

0.00 0.00 0.00 0.00 0.00 0.00 17.15 10.60 8.95



Sim 23 Sim 24 Sim 25 Sim 26 Sim 27 Sim 28 Sim 29 Sim 30 Sim 31100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00102.67 97.62 104.27 96.54 99.02 98.70 100.27 98.59 98.68101.90 93.42 102.65 95.36 95.55 99.43 100.57 100.12 98.45103.43 91.05 103.00 91.92 94.72 98.95 99.76 100.16 100.02103.21 91.05 103.97 92.42 92.36 99.53 98.47 96.05 96.92104.29 92.87 103.15 95.17 91.35 97.28 97.60 94.26 97.04102.74 93.51 101.52 92.47 92.06 94.55 97.52 94.63 96.21101.15 92.23 103.34 95.42 93.65 92.98 97.93 94.81 92.80104.98 91.86 103.56 98.51 93.91 96.19 96.80 93.90 93.32102.69 90.21 104.24 100.01 94.74 93.14 95.36 92.18 91.55104.72 90.33 102.28 97.61 93.67 90.94 97.09 90.76 91.51102.33 89.64 102.26 96.37 94.70 91.54 97.49 90.21 92.00

97.85 77.45 98.01 103.34 117.83 81.55 88.49 92.00 72.71

96.24 77.07 96.23 100.47 119.17 85.05 88.86 91.43 72.9997.01 78.83 99.44 98.04 118.07 86.43 88.32 91.23 72.4298.01 80.83 98.26 98.23 116.07 83.26 86.85 91.27 76.0498.07 81.82 100.25 97.72 115.67 81.91 86.61 90.08 79.0798.24 79.96 98.79 96.57 120.63 80.98 87.16 91.62 78.7899.76 78.89 96.10 93.53 124.76 79.68 81.43 91.81 78.67

101.66 77.02 96.68 93.51 126.55 79.32 80.27 91.56 80.89102.23 77.09 98.30 98.08 124.48 77.94 79.66 88.95 79.17106.04 79.92 101.54 101.36 125.11 80.38 81.23 87.64 80.43

99.78 84.48 99.29 108.78 99.71 88.83 92.35 85.80 88.10

0.00 0.00 0.00 3.78 0.00 0.00 0.00 0.00 0.00



Sim 32 Sim 33 Sim 34 Sim 35 Sim 36 Sim 37 Sim 38 Sim 39 Sim 40100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00

94.46 98.25 99.20 100.26 101.29 99.92 101.13 100.35 102.1195.23 97.97 101.23 100.60 102.89 99.07 102.76 100.92 103.1994.92 98.68 100.79 103.29 105.42 95.66 102.20 99.87 102.0096.77 97.90 102.24 105.41 108.45 98.60 104.68 98.78 98.6595.43 98.45 103.94 102.34 107.60 98.85 105.04 95.89 98.4395.21 99.00 103.53 97.83 106.67 98.31 107.48 95.99 99.0998.02 99.68 103.23 95.43 109.09 101.51 111.87 96.63 97.7097.42 99.02 103.49 98.68 112.32 105.33 110.18 96.30 97.1598.58 103.22 103.83 99.83 112.36 105.03 110.51 98.05 98.8296.07 102.55 102.49 102.90 110.62 105.84 113.13 98.61 99.8298.15 101.63 101.29 103.12 109.69 104.53 109.84 100.25 99.44

120.12 121.14 87.28 101.34 85.69 115.25 97.82 116.39 112.88

124.90 120.58 88.52 100.90 85.14 115.24 95.21 115.91 111.44123.65 121.23 92.67 100.86 84.79 114.58 92.86 117.99 112.52126.89 123.85 92.11 100.83 86.62 112.13 89.99 118.36 115.13127.68 117.39 94.65 99.34 85.86 109.63 90.50 119.26 117.35124.47 118.90 94.80 102.10 85.97 108.75 94.67 121.23 117.45125.99 117.96 93.75 101.90 86.96 108.83 93.85 119.44 113.60124.17 118.16 92.93 102.90 86.49 109.10 90.32 120.37 114.87124.65 118.12 95.03 100.38 87.22 114.64 89.42 122.58 113.43123.17 119.66 94.46 100.97 88.18 113.96 89.21 121.07 114.51

108.17 112.43 98.86 96.53 104.50 112.59 103.88 113.77 100.49

3.17 7.43 0.00 0.00 0.00 7.59 0.00 8.77 0.00




98.84 100.34 99.66 102.11 99.47 101.12 97.17 100.77 100.49100.09 99.92 99.64 101.23 103.54 101.85 92.68 105.13 101.16

99.86 100.03 96.87 102.36 104.96 99.23 91.32 105.55 102.30103.25 103.43 99.83 105.21 109.06 100.95 92.84 104.60 99.83103.07 104.55 101.44 101.61 110.30 100.22 94.07 105.04 101.27106.49 104.89 105.27 99.18 113.99 100.76 93.99 101.29 101.63100.22 106.62 106.48 96.53 118.44 106.01 92.61 104.17 101.58101.30 110.90 107.62 98.39 116.84 106.35 91.56 105.03 100.57100.54 112.25 107.82 97.02 120.04 106.73 94.85 105.64 99.00

98.60 113.37 109.84 99.14 121.58 106.62 96.00 105.84 98.18100.57 114.89 108.79 99.59 122.10 106.92 99.07 107.23 97.29

97.20 119.85 123.80 82.05 107.03 99.25 142.31 77.32 106.59

95.34 119.35 129.22 80.91 109.72 103.33 138.20 78.02 103.0397.68 118.53 127.96 80.58 109.10 106.31 139.39 74.96 103.4297.01 117.50 132.07 81.89 107.76 106.77 141.75 77.13 104.8996.72 117.69 131.35 82.91 108.69 108.06 138.61 74.88 104.0694.97 118.14 128.73 83.01 110.88 107.88 140.43 73.44 106.6495.12 124.34 124.63 83.92 115.83 108.22 142.04 71.66 106.7389.76 124.63 126.58 82.35 113.80 106.69 145.66 70.24 107.9888.50 120.35 125.42 82.91 112.55 107.47 143.18 71.60 110.0190.44 119.46 126.78 83.99 107.91 106.27 142.47 73.18 108.92

104.92 118.48 122.92 85.72 107.01 108.44 125.38 94.08 98.87

0.00 13.48 17.92 0.00 2.01 3.44 20.38 0.00 0.00




99.86 101.91 98.30 99.69 98.93 98.86 99.38 100.23 100.21100.32 101.58 97.20 103.55 101.35 98.92 98.29 97.12 102.82100.17 104.80 96.71 106.23 102.14 98.26 97.32 95.53 103.49102.99 107.67 98.61 103.75 103.59 96.91 96.82 95.37 103.86102.59 110.62 98.90 104.41 108.06 95.66 96.76 96.09 106.79101.23 106.75 100.59 103.79 106.01 93.61 98.65 97.19 110.08100.75 109.64 102.41 102.07 105.83 91.52 100.79 93.92 112.93105.59 109.45 102.26 96.95 106.82 90.88 96.46 93.49 113.31105.88 109.69 99.95 95.02 105.90 91.64 96.80 96.72 114.98105.92 109.94 98.52 93.30 103.42 88.62 95.76 95.87 116.11106.54 112.36 102.10 94.70 102.91 88.41 99.52 95.66 112.90119.34 121.57 125.45 96.48 140.77 105.01 119.01 80.08 123.42

119.40 119.20 123.51 96.90 146.38 105.04 116.99 78.58 126.98117.11 119.86 119.00 99.13 146.88 104.12 118.17 78.92 128.23118.26 114.52 122.55 98.69 146.70 106.52 119.46 80.69 128.68121.68 114.64 120.50 99.17 148.14 105.62 118.88 81.88 125.96117.84 112.96 122.05 97.47 147.20 104.02 117.89 82.36 124.59118.47 109.69 121.45 97.49 145.54 105.99 119.06 81.55 116.52119.74 110.91 121.16 97.47 146.01 106.03 121.51 81.24 113.35120.55 114.42 115.31 97.39 146.62 105.07 125.27 82.27 112.62118.45 109.71 114.20 99.47 144.28 105.75 124.81 80.36 114.95

116.07 114.14 112.87 103.87 113.83 92.89 104.60 85.49 124.99

11.07 9.14 7.87 0.00 8.83 0.00 0.00 0.00 19.99




146.73 102.98 102.91 82.22 86.96 74.18 114.09 101.35 102.84142.74 101.98 103.08 82.33 87.31 73.66 114.57 96.11 100.77137.64 103.80 99.90 80.53 90.73 71.69 115.85 97.78 103.18131.56 107.23 100.87 77.68 92.75 70.71 118.00 98.45 103.12129.87 102.63 99.16 77.23 92.03 70.67 121.84 96.23 105.15134.25 103.40 101.78 77.51 93.44 69.65 126.68 94.09 107.94135.39 103.94 102.47 78.34 91.50 72.84 131.77 91.11 107.16136.49 102.25 101.41 79.36 89.76 71.71 129.28 93.31 107.48137.23 102.61 103.04 79.07 88.30 70.82 131.04 95.42 110.35

124.87 98.97 100.86 88.13 87.72 88.48 116.07 103.16 103.72

19.87 0.00 0.00 0.00 0.00 0.00 11.07 0.00 0.00




99.61 98.61 104.41 97.20 103.50 97.93 103.01 100.40 98.2599.24 100.53 103.48 95.62 103.22 97.53 103.36 97.41 95.66

100.54 101.54 104.50 89.75 103.80 93.84 104.81 96.17 95.76100.23 101.18 104.99 90.08 106.15 93.04 101.18 95.81 98.21

99.44 100.87 103.66 90.49 107.77 91.98 100.38 93.92 102.5097.39 102.22 105.09 90.26 107.50 93.67 101.11 93.39 105.5198.93 104.07 104.17 92.93 107.82 95.67 99.43 96.46 109.1799.70 104.26 103.76 93.37 108.34 95.64 97.33 95.59 107.2795.37 104.85 104.80 92.07 104.44 95.18 98.99 96.68 107.5498.67 104.99 104.73 92.80 104.74 97.20 98.84 95.75 113.53

101.17 105.77 105.87 91.88 103.34 96.85 99.77 94.93 115.2876.22 104.46 106.06 101.59 121.64 61.72 80.00 89.41 104.47

77.09 105.19 106.18 104.15 120.06 61.58 81.94 88.47 108.7976.83 108.14 107.16 103.98 118.78 62.51 81.68 88.63 109.5078.14 111.20 107.76 104.97 120.77 64.15 78.99 87.66 111.2976.56 111.74 105.17 104.54 120.95 64.47 77.75 86.27 110.1578.00 113.35 109.80 105.61 121.33 62.60 77.81 87.16 110.3877.57 111.07 112.52 102.99 119.43 62.36 77.65 87.04 104.9977.37 109.23 112.50 105.77 123.66 61.56 78.62 86.62 105.2076.52 108.46 115.94 104.51 122.07 60.55 78.47 87.28 102.8077.32 106.93 118.49 102.75 119.09 60.32 75.67 87.04 104.77

90.79 100.83 107.30 95.85 114.26 83.70 91.38 88.49 108.63

0.00 0.00 2.30 0.00 9.26 0.00 0.00 0.00 3.63



Sim 86 Sim 87 Sim 88 Sim 89 Sim 90 Sim 91 Sim 92 Sim 93 Sim 94100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00102.35 101.05 98.78 98.68 99.84 97.61 97.66 101.15 100.15105.30 105.79 96.40 100.68 100.18 99.52 99.69 102.93 97.46103.87 105.34 99.25 99.81 98.94 98.85 97.65 101.53 97.20101.16 106.80 98.58 99.84 99.94 100.12 96.84 100.61 95.39100.21 103.43 98.59 108.19 98.40 100.00 96.81 98.52 96.37

99.26 107.41 98.80 105.64 96.42 99.44 96.85 96.68 93.9399.72 109.24 95.85 107.17 96.84 98.39 98.42 96.64 92.8298.98 108.24 95.66 108.11 94.76 97.48 99.03 92.97 96.4096.19 107.39 97.30 107.83 93.78 96.99 97.74 92.03 97.1996.37 109.55 95.11 107.19 92.64 99.96 97.39 94.44 94.8195.00 108.33 94.07 113.40 93.13 97.15 94.16 98.48 94.1386.32 119.28 84.20 128.87 95.52 108.50 102.16 117.11 148.90

87.62 114.76 85.68 129.44 92.55 110.14 105.29 123.04 152.8885.75 114.30 85.68 127.61 91.49 114.03 105.82 123.55 148.9787.77 114.22 84.83 126.41 91.10 113.45 105.41 127.71 147.0587.77 117.17 86.12 128.08 89.16 113.64 104.62 131.77 143.9889.87 117.61 87.91 129.49 90.23 114.56 105.31 127.96 143.3192.52 117.91 86.65 129.46 89.28 113.35 106.26 126.94 138.6993.46 120.63 84.49 126.80 89.06 113.60 107.64 128.91 135.7791.46 115.73 85.37 127.92 90.62 113.53 111.13 128.76 137.8494.18 117.32 83.94 129.15 89.07 116.52 111.00 129.04 137.63

101.74 111.07 93.54 126.52 93.19 105.23 96.53 105.89 121.22

0.00 6.07 0.00 21.52 0.00 0.23 0.00 0.89 16.22




161.79 88.81 89.15 108.05 99.13 86.00 91.01 141.92 124.88164.83 90.20 87.26 113.76 100.26 87.27 91.95 142.41 125.28167.60 88.76 89.36 120.95 99.42 85.22 93.60 140.60 123.78164.68 87.53 88.91 119.96 101.50 85.30 92.07 137.53 125.27158.42 87.62 89.51 117.27 106.28 83.72 90.00 131.36 124.46159.42 88.90 88.48 118.21 108.06 83.74 90.07 134.75 120.92168.61 92.93 86.41 117.30 107.40 85.53 90.06 135.98 118.63174.54 92.39 89.08 118.88 107.44 85.00 91.56 138.22 117.57171.28 88.44 89.21 117.50 107.47 86.94 92.64 138.70 117.03

132.72 94.34 97.74 107.72 100.25 91.40 91.19 120.47 110.96

27.72 0.00 0.00 2.72 0.00 0.00 0.00 15.47 5.96




125.24 112.44 113.90 102.48 117.15 107.83 96.94 84.30 105.78126.01 113.84 108.88 99.88 119.76 107.68 99.90 83.68 109.37124.58 115.95 112.24 100.03 118.32 104.45 99.61 84.43 114.35122.79 117.89 109.81 96.43 119.99 104.20 97.96 81.97 114.27124.38 120.93 110.05 98.60 118.83 103.42 100.86 78.20 114.23124.94 124.03 110.87 99.01 119.39 107.53 101.16 78.35 111.19127.87 124.61 105.99 99.63 120.74 106.15 103.06 79.37 117.08132.18 124.80 104.21 97.50 119.94 108.95 101.58 79.16 117.89133.01 127.12 103.79 95.06 123.24 108.80 102.77 75.78 120.23

116.71 117.88 105.96 102.38 106.71 104.09 98.32 88.97 112.05

11.71 12.88 0.96 0.00 1.71 0.00 0.00 0.00 7.05



Sim 113 Sim 114 Sim 115 Sim 116 Sim 117 Sim 118 Sim 119 Sim 120 Sim 121100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00101.33 98.58 99.31 99.53 102.35 102.10 98.67 100.67 100.85101.76 99.59 98.97 97.60 101.70 100.91 99.82 96.09 100.44102.30 100.43 98.77 95.75 103.57 96.96 99.24 95.96 99.20101.74 97.09 99.73 97.65 101.05 95.69 94.43 94.95 99.16103.34 95.61 98.66 96.36 100.25 95.42 97.57 95.87 99.08102.28 94.21 100.36 98.09 99.83 95.66 99.47 93.87 102.30100.91 97.34 97.04 95.87 98.45 97.03 99.15 90.50 100.95

97.80 97.94 102.22 99.61 98.27 95.05 100.71 89.83 102.74100.63 96.40 103.20 97.26 102.94 94.07 101.04 89.91 107.04102.46 96.10 103.33 99.82 104.37 94.41 101.22 92.91 106.35104.06 96.43 103.59 99.32 103.92 96.15 99.05 93.58 108.24134.71 110.28 90.27 139.01 80.04 86.06 118.74 103.83 95.07

134.41 110.15 87.71 140.67 77.84 87.28 121.12 102.44 97.78137.03 109.68 87.70 139.63 77.48 87.95 130.02 105.07 98.51140.89 108.55 85.19 135.78 79.79 87.82 129.81 103.77 96.29139.87 107.72 86.60 138.83 78.46 85.66 128.88 100.04 98.44139.36 107.49 87.29 138.17 75.32 82.68 124.26 100.68 98.06147.45 107.03 86.72 143.18 75.51 84.58 119.32 100.72 96.28151.85 108.65 85.50 143.88 76.04 84.83 120.56 97.56 97.80153.81 109.96 84.69 145.47 76.64 83.94 120.69 98.19 97.67159.97 114.97 86.82 146.05 77.30 87.76 123.21 96.49 99.12

116.23 108.55 90.14 123.24 90.54 93.98 105.80 105.04 107.94

11.23 3.55 0.00 18.24 0.00 0.00 0.80 0.04 2.94




99.90 102.94 99.72 101.00 101.01 98.84 101.64 102.62 105.56101.44 106.40 99.57 98.65 99.63 98.74 100.63 104.09 109.11100.15 107.00 99.98 98.23 101.14 98.53 104.34 102.31 108.03101.58 106.77 99.59 99.56 100.17 102.86 103.59 104.16 104.08100.55 106.51 99.00 97.96 100.64 100.46 104.40 103.99 102.99

98.14 105.17 99.60 97.13 99.78 98.00 103.25 101.98 101.1995.62 106.34 99.72 98.22 99.76 101.01 105.05 102.93 100.1494.69 105.81 100.03 102.86 99.79 99.45 104.07 99.77 99.8593.65 103.78 100.14 105.63 100.48 96.64 103.85 99.82 100.2192.64 103.33 102.76 103.19 99.10 93.67 102.68 99.16 97.1592.34 102.71 101.74 105.48 101.88 91.49 101.50 97.14 99.59

114.99 93.54 94.97 143.87 82.76 92.05 133.55 90.27 94.53

114.16 95.03 97.93 143.21 84.86 91.20 132.93 90.63 91.33116.55 99.17 94.61 145.24 82.15 93.72 129.60 90.83 93.36111.14 101.80 101.55 140.43 83.51 97.62 132.55 89.94 95.82112.64 96.96 99.96 141.62 86.12 97.33 132.53 89.38 96.09112.82 94.23 98.80 143.07 88.85 95.63 127.71 86.90 97.01114.04 95.86 95.88 143.85 91.22 95.08 124.07 86.73 98.28114.07 95.54 96.44 145.90 91.54 95.21 122.19 88.40 94.30111.58 93.85 98.07 146.31 90.06 98.57 121.81 89.43 91.75111.44 94.14 98.44 146.14 87.07 97.88 121.35 90.23 95.40

100.16 100.70 105.50 130.79 90.37 105.91 124.91 93.70 98.68

0.00 0.00 0.50 25.79 0.00 0.91 19.91 0.00 0.00



Sim 131 Sim 132 Sim 133 Sim 134 Sim 135 Sim 136 Sim 137 Sim 138 Sim 139100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00102.42 100.10 100.43 99.20 101.85 100.23 103.02 97.54 99.70103.21 99.48 99.21 99.09 98.83 100.35 103.67 97.82 96.53104.13 101.15 99.21 98.26 101.55 100.41 105.13 98.58 100.74100.20 98.86 96.10 95.11 99.01 104.35 100.99 99.82 103.51

98.53 100.28 94.99 96.68 96.56 104.23 104.07 99.83 106.3499.61 96.22 93.89 96.54 97.31 106.90 104.35 99.05 104.87

104.29 90.78 93.93 94.36 94.82 106.05 103.14 97.86 105.39101.81 93.23 96.83 93.88 95.21 111.45 99.65 99.22 103.38103.91 91.55 97.72 93.52 93.94 109.97 99.64 93.72 105.08101.36 86.60 98.45 94.64 93.71 108.75 100.81 94.03 105.57103.49 86.32 94.12 98.53 92.57 110.16 96.05 95.91 108.28

84.41 113.54 87.59 157.99 94.77 95.34 111.40 70.04 103.72

86.87 113.39 83.67 157.83 95.58 95.17 108.16 69.61 101.0386.93 115.12 83.66 156.25 94.37 93.63 106.54 69.20 98.9082.86 116.14 84.91 159.36 92.95 93.81 108.89 70.07 100.2284.96 120.16 84.01 156.61 92.02 93.90 107.78 69.94 102.0183.97 118.84 87.40 154.92 93.57 93.40 106.97 70.50 102.7984.78 120.71 87.55 151.05 93.33 95.80 104.65 68.59 103.2184.31 121.94 87.23 151.14 92.91 95.03 103.57 68.83 104.0783.36 118.18 86.87 153.12 92.78 96.53 104.27 68.98 104.8085.61 119.61 85.26 153.57 89.88 98.79 104.25 71.95 105.69

95.54 103.66 91.57 135.42 91.92 93.68 101.27 80.46 103.15

0.00 0.00 0.00 30.42 0.00 0.00 0.00 0.00 0.00




99.54 107.14 96.35 103.36 103.82 93.58 87.84 98.05 106.6799.74 107.72 95.72 102.84 100.01 92.91 90.56 99.44 105.64

100.33 109.65 96.31 101.87 96.65 91.21 89.88 99.35 101.80101.15 108.91 93.60 102.13 98.28 92.37 91.04 99.88 100.48

98.59 107.71 91.98 103.36 98.51 90.43 92.08 101.68 101.7298.59 105.66 91.14 105.49 99.82 89.23 94.65 101.06 101.19

125.39 89.86 98.97 150.43 187.60 116.22 102.98 104.97 101.32

125.86 89.28 97.85 152.00 183.02 118.00 102.33 107.08 99.13122.42 88.05 95.41 153.04 178.12 114.93 100.25 109.21 101.05121.98 89.24 93.28 154.19 172.43 116.41 99.75 108.34 102.38123.15 93.78 93.44 151.04 174.34 120.02 99.95 108.89 103.44122.10 92.94 95.23 147.60 178.55 118.96 98.53 106.95 105.97122.33 95.61 90.88 149.74 183.29 118.17 99.21 112.43 103.46120.44 96.10 88.33 151.50 179.82 117.65 99.11 110.01 101.23120.97 94.77 91.24 157.67 181.94 117.05 100.39 108.78 102.81121.78 98.03 91.21 160.89 184.34 115.74 97.95 110.57 99.47

109.74 98.20 95.27 129.91 155.15 103.58 101.57 107.01 101.76

4.74 0.00 0.00 24.91 50.15 0.00 0.00 2.01 0.00




117.04 90.96 142.54 108.46 79.30 87.70 80.13 89.13 98.72118.87 92.28 142.44 109.30 80.88 86.23 78.11 86.71 103.19120.90 93.17 145.40 109.50 79.00 89.03 76.55 87.42 107.02119.01 92.97 149.98 114.45 77.08 90.07 77.21 88.25 106.23119.20 94.46 144.32 114.30 77.02 91.00 78.24 89.76 108.19116.78 92.36 143.56 113.29 75.69 90.82 79.11 90.73 113.31118.34 91.86 148.71 112.82 76.89 93.29 77.33 90.51 111.16117.88 92.48 146.74 112.85 75.32 91.87 79.93 92.18 109.09116.99 91.50 150.40 114.62 75.41 91.84 79.46 90.90 108.32

112.45 99.00 119.05 109.14 93.11 90.58 82.99 92.58 98.76

7.45 0.00 14.05 4.14 0.00 0.00 0.00 0.00 0.00




98.19 99.47 100.66 97.85 102.71 99.34 96.56 100.71 101.0396.95 101.45 105.36 95.36 104.98 96.61 98.70 97.38 97.00

100.35 102.07 102.38 98.33 106.89 95.82 100.62 98.55 94.96100.60 102.07 103.27 100.78 107.33 98.66 99.70 94.29 93.08101.43 99.07 100.70 100.02 109.09 96.66 98.17 98.90 89.25100.52 100.58 103.91 103.44 111.29 96.39 95.64 100.68 90.18102.49 102.07 100.20 102.71 107.03 97.92 98.68 95.48 89.00102.05 104.53 104.53 101.44 108.79 99.90 98.26 96.57 91.66100.91 108.41 104.98 101.67 108.84 98.75 97.67 97.50 91.96

99.95 113.00 105.57 101.95 109.69 101.24 99.31 95.87 91.51101.62 115.03 107.16 99.20 110.93 102.27 98.92 96.23 91.64

97.21 130.19 78.17 102.27 104.79 99.79 124.64 123.70 93.56

94.79 132.30 79.21 98.28 104.26 99.94 124.72 127.18 92.9595.81 129.72 78.92 98.79 103.37 99.81 122.85 127.23 93.75

100.21 128.47 81.46 95.17 104.27 98.84 124.18 128.32 92.56101.90 123.71 81.23 96.83 103.49 101.52 126.25 129.79 93.09102.26 121.57 81.36 96.08 103.35 104.23 125.93 137.31 91.10103.17 117.68 78.39 94.50 102.73 105.09 125.92 133.22 88.18100.43 116.80 78.71 90.24 104.11 106.89 121.31 134.92 88.08102.55 118.90 77.56 89.06 106.64 110.35 119.79 135.72 87.60100.02 119.83 76.05 88.04 107.93 110.55 122.95 138.33 88.57

98.82 111.76 91.48 98.66 103.71 104.02 107.73 105.03 92.17

0.00 6.76 0.00 0.00 0.00 0.00 2.73 0.03 0.00




169.40 70.63 121.34 84.76 74.83 106.73 104.51 70.85 99.84171.31 71.16 119.92 87.75 75.15 109.81 103.23 71.96 102.15170.93 70.81 119.61 87.19 73.97 111.39 103.01 70.51 102.91175.62 69.46 116.77 88.98 75.32 109.34 98.91 69.57 103.70177.05 70.19 115.26 92.13 76.13 106.38 101.15 68.15 101.31176.62 70.36 113.24 91.40 74.96 105.82 98.22 67.56 99.45179.78 68.16 115.34 90.05 73.98 105.77 97.52 67.19 100.30176.09 69.28 116.83 91.23 71.87 106.86 95.89 65.88 99.89175.14 70.02 117.52 92.12 70.60 104.33 91.81 66.01 100.93

125.88 90.31 113.15 94.51 84.82 97.02 111.87 83.23 107.68

20.88 0.00 8.15 0.00 0.00 0.00 6.87 0.00 2.68




99.45 102.07 97.95 102.89 98.32 98.79 103.28 100.21 99.61100.81 107.48 96.67 105.64 100.04 99.94 102.70 102.27 97.43

99.72 105.38 99.27 106.51 98.85 100.18 101.48 103.83 94.74100.32 104.57 98.63 106.16 93.25 101.71 103.25 106.86 90.66100.23 100.11 101.03 106.96 91.87 103.95 106.05 105.93 90.14100.72 99.59 97.28 105.62 92.41 104.41 102.46 109.60 90.19104.50 98.17 93.31 103.90 92.51 104.87 105.14 111.67 94.23104.45 98.88 91.00 106.91 92.80 105.92 103.85 109.79 96.52103.40 101.79 91.01 106.64 94.78 104.14 102.03 107.27 95.44103.66 99.95 90.56 106.93 96.38 103.29 102.39 110.35 93.23103.49 102.58 93.21 107.77 97.95 103.79 103.07 106.58 95.09

96.20 90.11 123.80 98.61 82.09 76.11 112.89 114.98 149.54

91.60 91.37 123.83 96.95 80.01 76.90 118.16 112.86 150.4994.03 94.16 128.30 97.17 81.08 77.49 119.73 115.67 147.4393.56 92.10 127.11 98.63 81.96 76.28 118.74 119.39 148.3394.31 89.84 131.98 99.04 79.92 73.42 118.85 120.11 149.2791.54 92.03 135.48 100.38 79.48 75.16 119.37 120.87 151.6593.08 93.77 134.10 100.30 81.14 74.26 120.47 120.18 150.3492.79 90.20 130.33 100.85 84.32 73.12 119.06 124.61 146.4092.01 86.36 126.19 96.71 85.55 70.89 122.00 130.76 149.2494.11 85.74 130.35 94.59 85.69 69.60 122.84 130.50 149.20

98.33 94.60 102.88 99.51 89.94 86.69 115.44 107.78 126.20

0.00 0.00 0.00 0.00 0.00 0.00 10.44 2.78 21.20




108.75 73.79 93.50 133.05 122.39 87.99 100.64 104.30 107.20110.37 73.55 90.55 131.77 119.58 90.51 98.60 103.04 108.45108.12 73.28 90.22 133.86 120.77 92.18 97.19 107.74 108.48104.38 72.88 89.49 136.23 120.37 90.39 97.57 107.06 111.68103.87 73.11 88.76 133.18 121.87 91.60 101.02 104.39 109.44105.53 71.89 87.88 131.48 122.83 91.41 101.77 103.06 106.85105.35 71.50 88.27 130.32 125.58 94.31 103.29 99.44 107.27108.38 73.31 87.75 130.92 130.91 97.13 104.64 95.53 106.88110.69 72.47 87.46 133.53 130.35 96.47 103.14 95.12 109.05

114.60 87.11 87.51 127.60 114.25 95.58 95.69 105.67 99.15

9.60 0.00 0.00 22.60 9.25 0.00 0.00 0.67 0.00




99.40 100.86 98.20 102.70 101.08 100.21 99.69 99.92 96.6797.78 103.42 99.76 102.47 101.48 99.26 99.12 99.15 98.7398.70 103.65 101.81 101.72 104.40 100.11 98.44 101.86 96.5199.17 104.18 101.78 102.03 101.18 101.83 99.35 98.99 98.8496.89 108.18 99.84 102.96 98.99 104.32 95.96 97.18 98.8499.76 106.46 98.13 102.94 101.27 104.82 96.05 98.34 101.3898.37 105.75 97.76 97.87 98.80 108.57 95.32 100.09 99.4795.15 107.80 95.48 97.78 96.76 107.05 93.43 101.01 98.4393.70 108.28 93.35 99.09 94.93 110.63 93.60 99.54 101.1293.39 111.55 94.19 101.73 98.57 110.11 92.97 100.45 101.9992.85 111.04 94.76 103.90 100.26 110.90 93.89 101.07 102.09

125.03 137.58 94.54 114.97 90.40 88.56 81.32 98.81 119.50

127.24 140.26 92.97 113.63 86.08 86.43 81.22 98.02 119.06131.58 141.66 92.21 114.23 89.03 87.26 81.81 95.61 118.89129.87 143.78 91.94 112.69 87.67 85.45 83.56 93.86 115.89135.30 143.55 92.40 112.56 85.59 85.85 83.98 93.50 114.79134.91 144.40 92.82 110.61 85.96 87.27 83.82 93.15 115.32134.84 144.62 90.44 110.95 83.41 88.39 80.72 90.61 109.91135.44 150.24 91.94 110.73 83.75 88.93 82.65 88.29 109.60129.44 152.63 90.71 114.10 82.95 89.94 84.40 90.91 109.70129.06 156.76 92.59 114.02 83.20 89.22 83.35 90.98 113.42

108.63 123.71 96.63 108.37 97.54 98.68 86.44 90.91 104.44

3.63 18.71 0.00 3.37 0.00 0.00 0.00 0.00 0.00




97.46 101.98 95.96 97.47 99.70 100.90 100.26 101.26 100.4295.50 102.86 96.31 95.72 100.09 102.95 103.06 105.30 105.4795.96 101.65 94.64 92.42 103.19 105.21 102.59 105.96 107.9997.61 100.95 91.81 93.90 104.33 105.58 98.50 106.08 104.9898.72 100.37 91.55 91.56 106.88 103.51 99.16 106.73 103.4398.75 102.27 90.21 92.06 106.40 99.09 98.04 107.08 104.09

100.74 106.35 91.47 94.73 104.27 100.08 100.97 108.90 103.82100.21 106.18 92.28 92.51 101.12 102.49 101.01 109.12 100.23

99.20 106.97 89.40 93.79 97.99 104.53 100.96 107.21 99.5799.83 108.74 91.65 92.79 101.40 103.35 101.18 108.82 101.7699.21 108.50 91.27 95.56 101.79 104.60 101.76 110.14 100.6094.12 127.65 109.76 79.36 99.09 84.71 114.18 106.55 105.11

94.97 124.36 111.98 78.86 99.19 81.46 112.48 107.41 102.65100.00 123.83 110.77 80.17 99.79 82.38 111.92 107.30 105.54

98.17 122.79 111.21 82.17 97.83 83.48 111.90 108.19 103.8797.54 128.68 109.08 83.52 97.65 82.86 112.73 113.50 104.5095.43 127.32 111.06 84.03 93.71 80.95 114.05 114.73 103.8194.85 125.52 113.47 85.87 93.83 80.78 112.86 119.52 101.6293.85 125.42 112.70 83.77 93.81 82.86 117.33 116.80 98.5294.30 125.23 112.30 82.88 91.18 85.14 118.93 114.39 101.1396.45 128.89 113.61 80.89 90.92 85.50 121.19 114.22 99.62

89.68 115.77 100.14 85.20 94.50 83.56 112.87 108.90 100.00

0.00 10.77 0.00 0.00 0.00 0.00 7.87 3.90 0.00




99.99 101.55 99.88 103.73 98.93 105.12 96.54 97.48 100.20102.53 101.39 96.75 99.63 99.84 105.36 96.17 93.58 99.44101.57 103.03 94.25 102.66 96.38 103.08 95.64 93.95 99.86

99.82 104.86 94.88 103.81 96.42 102.28 94.99 93.34 98.3499.31 100.94 97.15 104.89 95.25 105.19 94.45 94.36 99.3695.58 99.23 94.97 106.60 95.63 102.92 94.02 93.54 99.0595.32 101.15 94.80 104.40 92.60 103.44 95.56 93.47 97.5995.50 100.60 95.95 104.08 92.73 100.70 95.59 93.42 98.9995.86 100.10 95.10 106.89 94.14 104.25 95.96 92.90 97.0692.84 101.91 94.96 107.93 93.69 101.60 97.23 90.78 99.9394.59 99.41 94.26 109.67 97.46 101.39 94.43 88.80 99.31

112.86 93.75 104.06 128.59 78.67 78.05 70.18 83.89 103.87

113.20 93.19 103.98 126.47 81.21 79.19 68.14 84.31 104.60114.47 92.44 101.07 122.53 80.98 78.31 68.27 84.08 104.25112.38 91.90 98.32 121.88 81.78 78.65 68.82 82.49 102.07114.52 89.54 100.76 120.25 81.98 75.03 67.85 82.63 103.55118.49 92.93 98.76 117.73 81.95 74.44 68.45 81.35 104.91118.29 91.19 97.77 115.93 81.10 76.80 67.57 80.81 105.26114.57 90.42 97.68 115.76 82.07 76.46 67.28 77.43 103.25115.71 91.62 94.06 111.92 82.50 74.25 66.62 74.98 103.36120.60 89.66 93.76 112.18 83.15 74.30 67.14 73.32 104.45

103.26 97.39 101.46 113.63 92.23 90.24 85.13 91.03 101.49

0.00 0.00 0.00 8.63 0.00 0.00 0.00 0.00 0.00




99.45 108.40 106.12 98.15 97.25 99.81 96.38 104.62 99.0599.75 107.69 108.73 96.74 98.66 99.18 96.55 107.23 98.1997.67 109.55 109.36 94.53 98.50 101.81 97.14 105.23 99.5298.90 106.56 106.41 96.23 99.21 100.04 96.58 106.34 103.2595.67 107.30 104.75 97.11 98.69 95.89 93.84 103.70 103.4096.36 107.28 108.16 99.32 99.54 93.52 95.16 104.69 106.36

105.90 103.54 134.67 84.55 91.84 86.89 107.20 118.72 95.06

106.57 102.31 132.80 84.75 94.08 85.51 110.35 121.71 91.48102.46 100.18 130.44 83.42 92.42 88.48 105.74 121.82 91.92101.45 100.12 129.40 82.67 92.80 85.41 105.91 120.89 90.83100.99 102.20 129.95 83.43 95.77 85.10 102.19 117.40 89.69102.58 105.11 129.49 82.64 98.33 86.53 99.10 116.79 88.88104.96 105.57 128.87 81.80 95.14 86.40 99.10 125.81 87.02104.83 108.59 129.35 80.90 97.63 87.53 96.49 122.40 87.82104.79 108.30 127.03 81.31 97.20 87.95 95.92 124.41 90.24106.09 107.95 125.68 82.65 93.13 90.80 93.90 125.93 89.43

96.28 110.00 119.90 92.24 94.40 87.79 96.24 113.93 99.78

0.00 5.00 14.90 0.00 0.00 0.00 0.00 8.93 0.00




95.53 97.37 100.02 98.12 100.82 97.34 100.40 100.09 98.5894.71 98.23 101.21 98.28 103.55 97.74 100.69 100.21 98.7393.89 95.62 99.80 97.81 102.03 96.64 97.94 95.17 104.0692.31 96.40 99.31 96.31 104.09 94.37 97.45 94.54 102.5893.07 96.72 103.91 95.13 103.18 94.94 97.24 94.00 99.9992.55 93.75 104.25 94.68 107.93 93.37 97.73 94.27 99.4391.29 95.53 103.16 92.41 106.50 92.30 97.11 95.99 102.8890.13 94.62 102.35 92.50 106.95 93.91 94.19 95.72 104.1390.79 96.51 102.22 90.96 104.96 94.13 94.67 96.13 101.8693.94 93.73 98.89 91.86 102.88 96.02 95.92 96.88 102.4692.94 94.05 98.49 93.57 103.25 98.17 94.04 97.72 102.49

104.47 75.69 112.26 128.75 94.12 99.19 64.09 115.72 141.98

103.97 77.66 116.94 127.15 95.78 103.48 65.89 116.59 147.89102.57 79.85 117.01 126.55 98.17 103.38 65.04 119.71 153.17102.50 80.68 115.54 128.46 95.55 104.02 62.87 119.06 155.89101.06 81.49 115.16 123.75 90.44 105.14 62.56 119.01 154.78104.14 80.89 110.53 122.10 91.31 99.44 63.74 116.20 155.85107.86 82.75 106.93 121.75 88.71 99.01 63.70 119.43 153.55104.33 80.80 103.35 120.67 91.09 97.98 62.84 122.21 153.11104.50 76.88 101.67 115.64 93.04 100.05 62.46 119.48 155.98104.58 74.01 103.05 111.35 92.68 100.68 62.67 117.79 156.34

96.15 84.65 104.76 107.44 94.32 91.60 81.16 99.66 135.80

0.00 0.00 0.00 2.44 0.00 0.00 0.00 0.00 30.80



Sim 248 Sim 249 Sim 250 Sim 251100.00 100.00 100.00 100.00103.34 100.54 99.71 98.92104.25 100.63 99.55 99.53103.56 103.14 98.91 99.13102.99 102.36 98.08 100.70104.12 101.26 100.08 101.43103.88 101.45 99.82 102.33103.24 100.36 99.94 100.88104.25 99.43 102.19 102.88105.05 100.05 100.40 106.37104.73 99.35 101.18 108.42105.31 101.99 99.89 104.53110.54 86.71 104.80 101.81

111.28 88.05 103.75 101.05110.09 85.54 105.54 100.47114.55 84.57 103.40 97.77117.34 83.54 101.54 96.21115.34 83.18 106.13 94.31117.80 82.38 107.97 95.89118.72 79.71 106.61 95.84120.13 80.43 107.12 97.16119.20 80.67 108.22 96.71

104.29 100.20 107.95 102.34

0.00 0.00 2.95 0.00



© Paul Wilmott 2001Box-Muller algorithm

Normal?#VALUE! Bins Mid pts. Freq.#VALUE! -2.5 #VALUE!#VALUE! -2.3 -2.4 #VALUE!#VALUE! -2.1 -2.2 #VALUE!#VALUE! -1.9 -2 #VALUE!#VALUE! -1.7 -1.8 #VALUE!#VALUE! -1.5 -1.6 #VALUE!#VALUE! -1.3 -1.4 #VALUE!#VALUE! -1.1 -1.2 #VALUE!#VALUE! -0.9 -1 #VALUE!#VALUE! -0.7 -0.8 #VALUE!#VALUE! -0.5 -0.6 #VALUE!#VALUE! -0.3 -0.4 #VALUE!

#VALUE! -0.1 -0.2 #VALUE!#VALUE! 0.1 0 #VALUE!#VALUE! 0.3 0.2 #VALUE!#VALUE! 0.5 0.4 #VALUE!#VALUE! 0.7 0.6 #VALUE!#VALUE! 0.9 0.8 #VALUE!#VALUE! 1.1 1 #VALUE!#VALUE! 1.3 1.2 #VALUE!#VALUE! 1.5 1.4 #VALUE!#VALUE! 1.7 1.6 #VALUE!#VALUE! 1.9 1.8 #VALUE!#VALUE! 2.1 2 #VALUE!

#VALUE! 2.3 2.2 #VALUE!#VALUE! 2.5 2.4 #VALUE!#VALUE! #VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!

#VALUE!#VALUE!

-3 -2 -1 0 1 2 30

2

4

6

8

10

12



#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!

#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!

#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!





#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!



© Paul Wilmott 2001Pricing interest rate products. Inputs in grey boxes

Spot rate 10% Time Sim 1 Sim 2 Sim 3 Sim 4Mean rate 8% 0 10.00% 10.00% 10.00% 10.00%

Reversion rate 0.2 0.01 10.07% 9.97% 9.96% 9.98%Volatility 0.01 0.02 10.08% 9.89% 10.00% 9.99%

0.03 10.05% 9.94% 9.98% 9.93%Strike 10% 0.04 9.98% 9.91% 10.10% 9.89%

Expiry 1 0.05 9.91% 9.89% 10.06% 9.86%0.06 9.87% 9.82% 10.08% 9.92%

Product? 0.0007 0.07 9.90% 9.80% 10.04% 10.00%0.08 9.80% 9.86% 10.03% 10.07%

Timestep 0.01 0.09 9.89% 9.89% 10.03% 10.15%0.1 9.97% 9.79% 10.08% 10.24%

0.11 10.04% 9.71% 10.02% 10.20%What does this 0.12 10.05% 9.69% 10.15% 10.20%spreadsheet calculate? 0.13 9.98% 9.67% 10.22% 10.20%

0.93 9.75% 9.65% 10.00% 10.38%How is this different from 0.94 9.64% 9.58% 9.90% 10.41%equity option valuation? 0.95 9.57% 9.39% 9.95% 10.32%

0.96 9.54% 9.37% 9.96% 10.30%0.97 9.59% 9.47% 10.01% 10.20%0.98 9.55% 9.53% 10.04% 10.26%0.99 9.57% 9.59% 9.96% 10.20%

1 9.48% 9.49% 9.88% 10.13%

Mean rate 9.89% 9.68% 10.20% 10.44%

Payoff 0.0000 0.0000 0.0000 0.0013

PV'd 0.0000 0.0000 0.0000 0.0012Mean 0



Sim 5 Sim 6 Sim 7 Sim 8 Sim 9 Sim 10 Sim 11 Sim 12 Sim 13 Sim 1410.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00%

9.94% 10.02% 9.91% 9.95% 9.97% 10.10% 9.91% 10.09% 9.97% 9.90%10.12% 9.94% 9.92% 9.99% 9.86% 10.01% 10.00% 10.10% 9.98% 9.92%10.07% 10.07% 9.89% 10.05% 9.97% 9.99% 9.95% 10.09% 9.93% 9.78%10.11% 10.17% 9.81% 9.97% 9.89% 9.97% 9.97% 10.04% 9.83% 9.70%10.07% 10.16% 9.88% 10.00% 9.83% 9.99% 9.93% 9.86% 9.92% 9.71%10.14% 10.08% 9.99% 10.01% 9.81% 10.02% 9.96% 9.82% 9.87% 9.70%10.06% 9.96% 9.87% 10.09% 9.88% 9.93% 9.85% 9.83% 9.86% 9.56%10.00% 10.05% 9.93% 10.10% 9.80% 9.95% 10.00% 9.88% 9.79% 9.58%

9.92% 10.09% 10.06% 10.12% 9.86% 9.91% 10.00% 9.82% 9.76% 9.61%9.89% 9.99% 10.04% 10.11% 9.72% 10.04% 10.04% 9.80% 9.84% 9.60%9.97% 10.07% 10.00% 9.98% 9.71% 10.08% 10.06% 9.97% 9.82% 9.49%9.95% 9.98% 9.98% 9.97% 9.73% 9.91% 9.99% 9.88% 9.77% 9.42%9.99% 9.92% 9.91% 10.03% 9.62% 9.88% 9.98% 9.79% 9.80% 9.41%9.50% 11.69% 9.90% 9.52% 9.36% 9.88% 9.37% 9.73% 9.82% 8.58%9.44% 11.64% 9.95% 9.57% 9.41% 9.99% 9.42% 9.75% 9.77% 8.58%9.48% 11.75% 9.94% 9.51% 9.54% 9.92% 9.44% 9.62% 9.74% 8.63%9.46% 11.72% 10.01% 9.63% 9.47% 9.92% 9.38% 9.61% 9.72% 8.69%9.34% 11.74% 9.98% 9.53% 9.45% 9.96% 9.48% 9.55% 9.68% 8.63%9.40% 11.78% 10.08% 9.66% 9.41% 9.87% 9.39% 9.49% 9.64% 8.61%9.52% 11.89% 10.13% 9.70% 9.38% 10.01% 9.35% 9.48% 9.68% 8.59%9.53% 11.85% 10.05% 9.85% 9.33% 9.88% 9.41% 9.50% 9.69% 8.56%

9.74% 10.88% 9.97% 9.56% 9.49% 9.75% 9.33% 9.75% 9.79% 8.93%

0.0000 0.0185 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.0000 0.0166 0 0 0 0 0 0 0 0



Sim 15 Sim 16 Sim 17 Sim 18 Sim 19 Sim 20 Sim 21 Sim 22 Sim 23 Sim 2410.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00%10.11% 9.92% 9.94% 9.94% 9.97% 9.95% 9.97% 9.99% 10.01% 9.84%10.04% 9.91% 10.00% 9.96% 9.98% 9.98% 10.02% 9.97% 9.99% 9.83%10.04% 10.03% 10.10% 9.90% 10.00% 10.00% 10.09% 9.83% 9.87% 9.78%10.07% 10.20% 10.20% 9.77% 9.93% 9.96% 10.02% 9.84% 9.77% 9.82%

9.99% 10.18% 10.21% 9.78% 9.84% 9.94% 10.05% 9.78% 9.66% 9.83%10.06% 10.25% 10.19% 9.62% 9.82% 10.02% 9.98% 9.75% 9.65% 9.80%

9.96% 10.32% 10.23% 9.59% 9.72% 10.05% 9.98% 9.65% 9.69% 9.73%9.87% 10.35% 10.07% 9.55% 9.73% 10.12% 9.94% 9.64% 9.46% 9.57%9.89% 10.40% 10.07% 9.40% 9.66% 10.19% 9.94% 9.69% 9.53% 9.61%9.98% 10.36% 10.01% 9.35% 9.69% 10.26% 9.99% 9.63% 9.69% 9.62%9.99% 10.35% 9.93% 9.39% 9.82% 10.27% 9.99% 9.53% 9.62% 9.55%

10.07% 10.32% 9.91% 9.33% 9.79% 10.16% 9.97% 9.69% 9.59% 9.55%10.12% 10.38% 9.90% 9.48% 9.86% 10.17% 9.83% 9.74% 9.70% 9.60%

9.60% 11.07% 9.91% 8.67% 9.54% 10.70% 10.12% 7.77% 10.45% 8.95%9.56% 11.04% 9.90% 8.70% 9.48% 10.62% 10.06% 7.80% 10.47% 8.98%9.49% 10.91% 9.90% 8.69% 9.39% 10.56% 9.91% 7.78% 10.51% 9.04%9.50% 10.87% 10.01% 8.78% 9.38% 10.50% 9.91% 7.65% 10.43% 9.17%9.36% 10.88% 10.01% 8.80% 9.47% 10.45% 9.85% 7.74% 10.31% 9.32%9.38% 10.74% 10.07% 8.77% 9.49% 10.36% 9.77% 7.69% 10.25% 9.33%9.47% 10.76% 10.04% 8.70% 9.38% 10.44% 9.89% 7.78% 10.21% 9.30%9.45% 10.80% 9.95% 8.70% 9.26% 10.30% 9.82% 7.80% 10.30% 9.34%

9.49% 10.54% 10.16% 9.08% 9.70% 10.31% 9.87% 8.62% 9.99% 9.24%

0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0.01 0 0 0 0 0 0 0 0



Sim 25 Sim 26 Sim 27 Sim 28 Sim 29 Sim 30 Sim 31 Sim 32 Sim 33 Sim 3410.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00%10.07% 10.13% 9.91% 9.97% 9.96% 10.06% 9.96% 10.09% 9.95% 9.90%

9.99% 10.21% 9.96% 9.98% 10.01% 10.01% 9.98% 10.14% 10.00% 9.87%10.02% 10.22% 10.08% 9.90% 10.07% 9.85% 9.99% 10.18% 10.03% 9.71%

9.96% 10.23% 10.07% 10.00% 10.02% 9.82% 10.00% 10.16% 10.09% 9.70%9.87% 10.18% 10.06% 10.11% 9.90% 9.74% 10.01% 10.22% 10.06% 9.59%9.89% 10.24% 10.08% 10.09% 9.92% 9.85% 10.01% 10.36% 10.12% 9.59%9.85% 10.31% 9.99% 10.12% 9.88% 9.90% 9.93% 10.36% 10.15% 9.51%9.78% 10.21% 10.05% 10.12% 9.93% 9.80% 9.94% 10.38% 10.13% 9.44%9.80% 10.17% 10.10% 10.02% 9.94% 9.74% 10.02% 10.27% 10.18% 9.51%9.79% 10.22% 10.10% 9.95% 9.88% 9.68% 10.14% 10.34% 10.08% 9.64%9.81% 10.24% 10.03% 10.03% 9.86% 9.64% 10.16% 10.27% 10.17% 9.61%9.86% 10.28% 10.02% 9.97% 10.04% 9.75% 10.30% 10.30% 10.22% 9.62%9.77% 10.28% 9.87% 10.02% 10.08% 9.80% 10.28% 10.32% 10.14% 9.63%8.90% 8.92% 9.32% 9.61% 9.73% 9.22% 10.57% 9.41% 9.95% 9.25%8.92% 8.91% 9.37% 9.64% 9.85% 9.29% 10.58% 9.40% 9.88% 9.18%8.99% 8.95% 9.53% 9.60% 9.95% 9.34% 10.50% 9.39% 9.91% 9.16%9.11% 8.94% 9.49% 9.64% 9.95% 9.35% 10.46% 9.43% 9.87% 9.30%9.13% 8.96% 9.47% 9.70% 9.93% 9.48% 10.47% 9.41% 9.97% 9.39%9.18% 8.99% 9.39% 9.65% 9.96% 9.47% 10.45% 9.41% 10.02% 9.29%9.13% 8.93% 9.49% 9.65% 9.93% 9.43% 10.36% 9.38% 10.11% 9.35%9.05% 8.88% 9.58% 9.64% 10.02% 9.37% 10.29% 9.48% 10.19% 9.24%

9.35% 9.70% 9.59% 9.78% 9.89% 9.73% 10.36% 9.64% 9.88% 9.43%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.90% 9.96% 10.12% 9.91% 9.96% 10.06% 9.89% 10.02% 10.09% 9.95%9.94% 9.98% 10.16% 9.94% 9.99% 9.98% 9.82% 10.01% 9.95% 9.99%9.70% 10.03% 10.09% 9.89% 10.00% 9.95% 9.74% 9.95% 9.91% 10.06%9.68% 9.98% 10.20% 10.09% 9.92% 9.94% 9.74% 9.95% 10.01% 9.93%9.61% 9.91% 10.28% 10.03% 9.89% 9.94% 9.81% 9.98% 10.08% 10.07%9.57% 9.90% 10.40% 9.99% 9.90% 9.90% 9.73% 10.13% 9.97% 10.02%9.62% 9.75% 10.44% 9.96% 9.92% 9.83% 9.68% 10.21% 9.88% 9.96%9.66% 9.68% 10.40% 10.02% 10.04% 9.94% 9.57% 10.26% 9.78% 10.01%9.60% 9.69% 10.48% 10.13% 9.99% 10.08% 9.52% 10.20% 9.81% 10.04%9.44% 9.66% 10.50% 10.07% 10.03% 10.09% 9.43% 10.18% 9.78% 10.10%9.36% 9.68% 10.41% 10.07% 10.06% 10.08% 9.33% 10.05% 9.67% 9.99%9.34% 9.72% 10.37% 9.97% 10.09% 10.09% 9.24% 10.08% 9.69% 9.99%9.38% 9.77% 10.43% 10.03% 10.11% 10.13% 9.24% 10.11% 9.60% 9.97%9.67% 9.31% 9.56% 9.12% 10.36% 9.45% 9.51% 10.44% 8.69% 9.78%9.71% 9.30% 9.55% 9.19% 10.26% 9.42% 9.53% 10.36% 8.66% 9.82%9.64% 9.19% 9.57% 9.09% 10.24% 9.44% 9.46% 10.27% 8.66% 9.86%9.71% 9.22% 9.72% 9.07% 10.18% 9.37% 9.48% 10.16% 8.56% 9.77%9.66% 9.13% 9.75% 9.06% 10.20% 9.36% 9.41% 10.07% 8.71% 9.81%9.65% 9.15% 9.83% 9.05% 10.24% 9.51% 9.28% 10.08% 8.79% 9.74%9.67% 9.23% 9.82% 9.00% 10.35% 9.55% 9.47% 10.00% 8.74% 9.72%9.73% 9.18% 9.90% 8.93% 10.29% 9.51% 9.58% 9.98% 8.62% 9.78%

9.58% 9.43% 9.85% 9.39% 9.96% 9.76% 9.25% 10.50% 9.16% 9.77%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.91% 10.01% 10.00% 9.96% 10.12% 9.98% 9.96% 9.95% 10.04% 10.00%9.84% 10.00% 10.05% 10.10% 10.09% 10.05% 9.92% 9.96% 10.12% 10.09%9.90% 9.99% 10.04% 10.02% 9.91% 10.01% 9.84% 9.99% 10.02% 10.02%9.77% 10.01% 10.10% 9.90% 9.99% 10.01% 9.89% 9.85% 10.06% 9.84%9.82% 10.05% 10.22% 9.77% 9.91% 9.96% 9.94% 9.80% 10.08% 9.84%9.81% 10.02% 10.21% 9.79% 9.90% 9.93% 9.89% 9.91% 10.10% 9.90%9.80% 10.07% 10.20% 9.72% 9.88% 9.96% 9.96% 9.78% 10.07% 9.71%9.79% 10.08% 10.20% 9.61% 9.98% 9.89% 10.09% 9.74% 10.16% 9.73%9.69% 10.08% 10.27% 9.59% 9.90% 9.87% 10.13% 9.62% 10.06% 9.71%9.56% 10.13% 10.18% 9.65% 9.82% 9.89% 10.12% 9.59% 10.05% 9.69%9.57% 10.14% 10.23% 9.63% 9.84% 9.90% 9.90% 9.64% 10.09% 9.67%9.51% 10.27% 10.21% 9.62% 9.87% 9.91% 9.89% 9.59% 10.03% 9.51%9.45% 10.27% 10.11% 9.57% 9.70% 10.01% 9.83% 9.60% 10.03% 9.47%9.59% 9.21% 9.49% 9.39% 9.51% 8.21% 10.55% 9.17% 9.76% 9.53%9.46% 9.21% 9.52% 9.26% 9.46% 8.18% 10.52% 9.20% 9.70% 9.54%9.48% 9.26% 9.57% 9.32% 9.37% 8.19% 10.57% 9.26% 9.56% 9.64%9.47% 9.19% 9.60% 9.42% 9.48% 8.23% 10.56% 9.28% 9.57% 9.56%9.47% 9.18% 9.69% 9.36% 9.43% 8.19% 10.47% 9.21% 9.48% 9.47%9.46% 9.20% 9.73% 9.35% 9.48% 8.29% 10.45% 9.20% 9.62% 9.50%9.56% 9.27% 9.69% 9.42% 9.41% 8.28% 10.35% 9.30% 9.63% 9.56%9.58% 9.11% 9.67% 9.43% 9.38% 8.16% 10.38% 9.26% 9.76% 9.53%

9.76% 9.87% 9.76% 9.28% 9.61% 9.33% 10.23% 9.49% 9.74% 9.61%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0



Sim 55 Sim 56 Sim 57 Sim 58 Sim 59 Sim 60 Sim 61 Sim 62 Sim 63 Sim 6410.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00%10.14% 10.01% 9.97% 9.99% 10.03% 9.99% 10.07% 10.02% 9.97% 9.97%10.08% 10.05% 10.03% 9.92% 9.95% 10.15% 10.02% 10.07% 9.92% 10.01%10.04% 9.99% 10.11% 9.85% 9.96% 10.06% 10.08% 10.19% 9.97% 10.03%10.10% 10.06% 10.09% 9.92% 10.06% 10.03% 9.99% 10.16% 10.00% 10.01%10.04% 10.15% 10.10% 9.94% 10.09% 10.07% 9.91% 10.19% 9.88% 10.01%10.06% 10.15% 10.16% 9.87% 10.04% 10.04% 10.00% 10.05% 9.89% 10.04%10.11% 10.22% 10.13% 9.78% 10.08% 10.00% 10.07% 10.03% 9.85% 10.16%10.11% 10.23% 10.16% 9.77% 10.02% 9.89% 10.16% 10.06% 9.89% 10.10%10.17% 10.22% 10.20% 9.85% 10.06% 9.77% 10.20% 10.10% 9.88% 10.13%10.17% 10.08% 10.13% 9.90% 10.12% 9.88% 10.21% 10.11% 9.89% 10.11%10.18% 10.07% 10.01% 9.80% 10.16% 9.84% 10.17% 10.10% 9.91% 10.08%10.21% 9.98% 9.88% 9.87% 10.13% 9.89% 10.17% 10.15% 10.08% 10.10%10.05% 9.98% 9.82% 9.89% 10.12% 9.86% 10.19% 10.23% 10.12% 10.13%

9.55% 9.71% 9.89% 9.99% 8.98% 10.20% 9.79% 10.38% 9.45% 9.55%9.51% 9.71% 10.01% 9.96% 8.90% 10.21% 9.81% 10.50% 9.40% 9.58%9.36% 9.73% 10.05% 9.92% 8.94% 10.27% 9.77% 10.48% 9.30% 9.63%9.29% 9.73% 10.01% 9.93% 8.91% 10.28% 9.76% 10.50% 9.24% 9.54%9.35% 9.59% 10.07% 9.97% 8.87% 10.28% 9.89% 10.51% 9.24% 9.61%9.36% 9.57% 10.16% 9.96% 8.90% 10.31% 9.92% 10.57% 9.31% 9.69%9.38% 9.55% 10.11% 10.06% 8.85% 10.24% 9.92% 10.50% 9.36% 9.67%9.36% 9.50% 10.00% 10.04% 8.84% 10.21% 10.01% 10.49% 9.36% 9.59%

9.73% 9.70% 9.79% 10.15% 9.61% 10.29% 10.04% 10.40% 9.69% 10.02%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.94% 10.01% 9.93% 9.91% 9.99% 9.99% 9.94% 10.00% 10.02% 9.91%9.90% 9.91% 9.93% 9.99% 10.02% 9.91% 9.97% 9.97% 10.14% 9.96%9.92% 9.94% 9.89% 10.00% 10.03% 9.91% 9.98% 9.98% 10.06% 9.85%

10.04% 9.89% 10.02% 9.92% 10.03% 9.86% 10.03% 9.85% 9.96% 9.82%10.03% 9.88% 10.11% 9.98% 9.98% 9.87% 10.06% 9.74% 9.96% 9.84%10.01% 9.93% 10.14% 9.94% 9.92% 9.93% 10.06% 9.80% 9.95% 9.86%10.04% 9.97% 10.10% 9.86% 9.96% 9.87% 10.11% 9.84% 10.12% 9.85%10.09% 10.04% 10.08% 9.90% 10.14% 9.79% 10.23% 9.87% 9.97% 9.93%10.06% 9.95% 10.10% 9.98% 10.10% 9.78% 10.18% 9.87% 9.97% 10.04%10.09% 9.88% 10.08% 9.97% 10.18% 9.77% 10.23% 9.84% 9.88% 9.88%10.05% 9.77% 10.20% 9.94% 10.12% 9.73% 10.30% 9.82% 9.87% 9.92%10.06% 9.73% 10.19% 9.88% 10.08% 9.76% 10.28% 9.80% 9.88% 9.86%10.09% 9.69% 10.26% 9.82% 9.94% 9.78% 10.27% 9.70% 9.90% 9.96%

9.83% 9.53% 9.92% 9.51% 9.69% 8.54% 9.89% 8.17% 10.01% 8.77%9.76% 9.52% 10.01% 9.34% 9.74% 8.52% 9.87% 8.13% 10.13% 8.80%9.81% 9.51% 10.09% 9.29% 9.84% 8.56% 9.88% 8.15% 10.13% 8.83%9.82% 9.48% 10.08% 9.44% 9.78% 8.60% 9.85% 8.00% 10.12% 8.98%9.92% 9.57% 9.95% 9.36% 9.87% 8.55% 9.83% 8.16% 10.08% 8.95%9.92% 9.45% 9.88% 9.43% 9.83% 8.53% 9.92% 8.14% 10.09% 8.99%9.77% 9.63% 10.00% 9.51% 9.84% 8.60% 9.94% 8.08% 10.09% 9.01%9.80% 9.65% 10.12% 9.62% 9.99% 8.54% 9.78% 7.97% 10.12% 8.95%

9.86% 9.45% 10.15% 9.75% 9.75% 9.16% 9.99% 8.96% 10.08% 9.39%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.90% 9.99% 9.90% 10.01% 9.96% 9.85% 10.01% 9.91% 9.98% 10.00%9.92% 10.00% 9.87% 10.09% 10.00% 9.86% 10.00% 9.88% 10.02% 9.90%9.91% 10.13% 9.88% 10.09% 9.92% 9.83% 10.02% 9.79% 9.99% 9.88%9.82% 10.19% 9.84% 10.06% 9.95% 9.87% 10.03% 9.72% 9.96% 9.88%9.78% 10.14% 9.72% 9.96% 9.88% 9.74% 10.09% 9.86% 9.94% 9.83%9.78% 10.12% 9.76% 9.96% 10.01% 9.71% 10.08% 9.83% 9.95% 9.83%9.67% 10.15% 9.78% 9.90% 10.04% 9.63% 9.96% 9.74% 10.03% 9.93%9.57% 10.22% 9.71% 9.91% 10.00% 9.59% 10.03% 9.66% 10.07% 9.83%9.45% 10.15% 9.71% 9.92% 9.95% 9.60% 10.07% 9.61% 10.11% 9.93%9.48% 10.22% 9.65% 9.94% 9.92% 9.55% 9.99% 9.76% 10.07% 9.97%9.50% 10.26% 9.57% 9.73% 9.92% 9.53% 9.96% 9.71% 10.00% 9.91%9.60% 10.28% 9.50% 9.80% 9.97% 9.51% 10.10% 9.69% 10.00% 9.96%9.57% 10.21% 9.58% 9.81% 9.85% 9.57% 10.08% 9.73% 10.04% 10.02%8.82% 9.68% 9.91% 9.75% 10.21% 8.62% 10.11% 9.63% 10.40% 9.40%8.88% 9.71% 9.91% 9.72% 10.09% 8.66% 10.22% 9.58% 10.38% 9.39%8.94% 9.65% 9.96% 9.66% 10.00% 8.71% 10.32% 9.77% 10.40% 9.46%8.95% 9.80% 10.00% 9.69% 9.97% 8.64% 10.36% 9.83% 10.33% 9.30%8.91% 9.92% 9.96% 9.67% 9.94% 8.62% 10.26% 9.75% 10.32% 9.34%8.87% 9.77% 9.93% 9.76% 9.90% 8.68% 10.21% 9.70% 10.31% 9.35%8.87% 9.90% 10.01% 9.80% 9.88% 8.70% 10.13% 9.63% 10.26% 9.36%8.89% 9.88% 10.02% 9.84% 9.83% 8.63% 10.16% 9.67% 10.29% 9.39%

9.31% 10.09% 9.75% 9.69% 9.91% 9.08% 10.18% 9.52% 10.29% 9.95%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.96% 9.97% 9.98% 9.90% 10.07% 10.03% 9.91% 10.07% 10.20% 10.08%9.98% 9.90% 9.96% 9.95% 10.04% 10.01% 9.91% 10.11% 10.07% 10.09%

10.04% 9.99% 9.92% 9.89% 10.03% 10.11% 9.92% 10.21% 10.00% 9.94%10.06% 10.01% 9.81% 9.89% 10.04% 10.16% 9.91% 10.14% 9.90% 9.93%10.03% 9.95% 9.75% 9.88% 9.96% 10.14% 9.90% 10.09% 9.90% 9.93%10.05% 9.88% 9.73% 9.86% 9.98% 10.18% 9.95% 10.04% 9.82% 9.93%

9.97% 9.87% 9.73% 9.86% 9.94% 10.14% 9.87% 10.14% 9.73% 9.96%10.10% 9.87% 9.78% 9.82% 10.03% 10.11% 9.85% 10.21% 9.75% 9.93%10.16% 9.93% 9.82% 9.90% 10.05% 10.12% 9.89% 10.23% 9.82% 9.94%10.12% 9.92% 9.77% 9.78% 10.02% 10.02% 9.91% 10.31% 9.73% 9.90%10.13% 9.96% 9.81% 9.79% 10.07% 10.15% 9.82% 10.35% 9.59% 9.79%10.08% 9.88% 9.73% 9.80% 10.11% 10.18% 9.98% 10.32% 9.65% 9.76%10.07% 9.73% 9.77% 9.81% 10.19% 10.31% 9.96% 10.35% 9.84% 9.74%10.03% 8.40% 10.05% 9.84% 9.68% 9.71% 10.11% 9.73% 8.79% 10.58%

9.93% 8.33% 10.04% 9.83% 9.71% 9.65% 10.18% 9.62% 8.68% 10.60%10.00% 8.38% 10.10% 9.71% 9.66% 9.80% 10.21% 9.68% 8.77% 10.63%

9.99% 8.42% 10.20% 9.83% 9.71% 9.90% 10.23% 9.75% 8.77% 10.54%10.12% 8.38% 10.19% 9.90% 9.62% 9.94% 10.19% 9.82% 8.81% 10.55%10.20% 8.36% 10.22% 9.84% 9.64% 9.85% 10.25% 9.92% 8.89% 10.57%10.15% 8.32% 10.16% 9.79% 9.64% 9.85% 10.27% 9.94% 8.81% 10.48%10.15% 8.24% 10.10% 9.89% 9.81% 9.83% 10.27% 9.86% 8.77% 10.47%

9.88% 8.96% 9.79% 9.74% 9.88% 10.09% 10.12% 10.04% 9.26% 10.10%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.88% 10.09% 9.91% 9.95% 9.99% 10.11% 9.96% 9.92% 10.04% 10.09%9.79% 10.04% 9.99% 9.96% 9.89% 10.23% 9.99% 9.91% 9.95% 10.12%9.74% 10.03% 9.87% 9.95% 9.92% 10.18% 10.06% 9.89% 10.04% 10.26%9.81% 10.06% 9.89% 9.96% 9.92% 10.13% 9.93% 9.93% 9.99% 10.27%9.77% 10.21% 9.93% 9.95% 9.96% 10.20% 9.96% 9.89% 9.87% 10.31%9.73% 10.30% 9.92% 9.89% 9.93% 10.17% 10.00% 9.81% 9.75% 10.26%9.85% 10.26% 9.95% 9.96% 9.93% 10.16% 10.02% 9.91% 9.78% 10.14%9.94% 10.35% 9.88% 9.94% 9.91% 10.15% 10.05% 9.90% 9.87% 10.09%9.97% 10.30% 9.83% 10.00% 9.92% 10.17% 10.08% 9.87% 9.87% 10.10%9.95% 10.11% 9.84% 9.89% 9.96% 10.19% 10.04% 9.86% 9.76% 10.11%9.98% 10.03% 9.78% 9.93% 9.89% 10.23% 9.95% 9.83% 9.72% 10.06%

10.05% 9.91% 9.83% 9.93% 9.88% 10.20% 9.82% 9.87% 9.66% 10.08%10.08% 9.86% 9.80% 9.79% 9.97% 10.23% 9.69% 9.74% 9.73% 10.10%10.18% 9.59% 8.96% 11.34% 9.80% 8.27% 8.78% 9.74% 9.13% 9.05%10.01% 9.56% 9.02% 11.34% 9.72% 8.40% 8.79% 9.75% 9.00% 9.12%10.09% 9.60% 9.06% 11.41% 9.75% 8.39% 8.70% 9.57% 8.94% 9.10%10.20% 9.55% 9.08% 11.34% 9.62% 8.57% 8.78% 9.64% 8.94% 9.08%10.19% 9.59% 9.13% 11.31% 9.55% 8.48% 8.78% 9.67% 9.02% 9.04%10.25% 9.63% 9.05% 11.29% 9.57% 8.53% 8.75% 9.67% 9.07% 8.99%10.23% 9.66% 8.98% 11.26% 9.57% 8.53% 8.67% 9.75% 9.09% 9.15%10.30% 9.77% 8.95% 11.20% 9.52% 8.56% 8.85% 9.70% 9.09% 9.18%

9.92% 9.76% 9.61% 10.58% 9.74% 9.26% 9.33% 9.25% 9.39% 9.42%

0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0.01 0 0 0 0 0 0




9.86% 10.03% 10.10% 9.96% 9.99% 10.09% 9.98% 10.03% 9.92% 9.93%9.85% 10.04% 10.09% 10.03% 10.02% 10.12% 9.98% 10.08% 9.97% 9.90%9.77% 10.05% 10.06% 10.07% 10.04% 10.15% 9.88% 10.00% 9.88% 9.77%9.71% 9.99% 10.04% 10.13% 10.00% 10.20% 9.86% 10.01% 9.84% 9.82%9.70% 10.01% 10.12% 10.07% 10.03% 10.17% 9.93% 9.90% 9.90% 9.79%9.77% 10.01% 10.06% 10.14% 9.97% 10.12% 9.90% 9.87% 9.90% 9.75%9.73% 10.10% 10.07% 10.18% 9.92% 10.18% 9.75% 9.73% 9.88% 9.72%9.85% 10.23% 10.13% 10.22% 9.89% 10.14% 9.77% 9.73% 9.94% 9.65%9.78% 10.22% 10.10% 10.17% 9.91% 10.13% 9.71% 9.69% 9.93% 9.75%9.72% 10.21% 10.12% 10.17% 9.99% 10.24% 9.75% 9.70% 10.00% 9.70%9.60% 10.26% 10.09% 10.11% 9.88% 10.26% 9.82% 9.64% 9.98% 9.74%9.61% 10.24% 10.10% 10.06% 9.86% 10.14% 9.79% 9.67% 9.86% 9.68%9.70% 10.20% 10.17% 10.00% 9.85% 10.13% 9.74% 9.74% 9.81% 9.69%9.00% 10.61% 9.35% 9.81% 10.03% 9.31% 10.35% 9.63% 10.02% 10.13%8.98% 10.59% 9.35% 9.78% 10.02% 9.30% 10.24% 9.67% 9.93% 10.08%9.00% 10.62% 9.35% 9.72% 10.05% 9.17% 10.31% 9.65% 9.91% 9.95%8.86% 10.66% 9.36% 9.74% 10.08% 9.37% 10.26% 9.68% 9.88% 10.01%8.82% 10.61% 9.40% 9.73% 10.14% 9.36% 10.38% 9.80% 9.86% 10.04%8.83% 10.53% 9.41% 9.74% 10.13% 9.39% 10.38% 9.72% 9.84% 10.18%8.85% 10.57% 9.42% 9.73% 10.09% 9.41% 10.32% 9.75% 9.77% 10.21%8.82% 10.62% 9.47% 9.82% 10.18% 9.43% 10.29% 9.81% 9.83% 10.22%

9.37% 10.28% 9.82% 9.92% 9.82% 10.01% 9.97% 9.59% 10.00% 9.74%

0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0.01 0 0 0 0 0 0 0 0




9.95% 9.95% 9.96% 9.92% 9.93% 9.94% 9.96% 10.02% 9.90% 10.03%9.84% 9.85% 9.85% 9.98% 9.84% 9.84% 9.86% 10.08% 9.84% 9.99%9.89% 9.90% 9.91% 10.01% 9.63% 9.85% 9.75% 10.11% 9.86% 9.92%9.89% 9.91% 9.97% 10.02% 9.66% 9.79% 9.69% 10.23% 9.78% 9.89%9.87% 9.85% 9.89% 9.96% 9.62% 9.83% 9.59% 10.24% 9.70% 9.85%9.97% 9.77% 9.88% 10.05% 9.63% 9.94% 9.68% 10.31% 9.59% 9.84%

10.02% 9.84% 9.84% 10.12% 9.75% 9.91% 9.67% 10.32% 9.70% 9.76%9.97% 9.90% 9.80% 10.03% 9.67% 9.97% 9.68% 10.33% 9.63% 9.76%9.87% 9.97% 9.77% 9.98% 9.80% 9.98% 9.71% 10.32% 9.64% 9.73%9.93% 10.08% 9.77% 9.85% 9.85% 10.02% 9.81% 10.29% 9.76% 9.90%9.96% 10.01% 9.77% 9.85% 9.90% 10.01% 9.65% 10.37% 9.81% 9.82%

10.00% 10.03% 9.82% 9.82% 9.82% 10.00% 9.58% 10.38% 9.73% 9.88%9.93% 9.95% 9.77% 9.86% 9.81% 9.97% 9.71% 10.40% 9.70% 9.83%

10.46% 9.80% 9.60% 9.89% 9.07% 9.12% 9.57% 9.47% 8.03% 9.09%10.45% 9.84% 9.66% 9.76% 9.23% 9.17% 9.59% 9.40% 8.13% 9.25%10.44% 9.79% 9.65% 9.75% 9.16% 9.11% 9.61% 9.41% 8.18% 9.29%10.47% 9.87% 9.57% 9.82% 9.11% 9.12% 9.61% 9.33% 8.10% 9.28%10.46% 9.80% 9.56% 9.77% 9.21% 9.17% 9.56% 9.37% 8.07% 9.26%10.41% 9.73% 9.65% 9.69% 9.23% 9.21% 9.56% 9.30% 8.01% 9.33%10.33% 9.71% 9.68% 9.72% 9.08% 9.25% 9.59% 9.44% 7.91% 9.39%10.19% 9.64% 9.57% 9.75% 9.12% 9.43% 9.61% 9.47% 7.86% 9.34%

10.35% 9.75% 9.75% 9.90% 9.49% 9.83% 9.85% 10.04% 8.77% 9.52%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.81% 10.02% 10.07% 10.02% 10.01% 10.02% 10.08% 10.04% 10.03% 9.98%9.77% 9.85% 9.96% 10.12% 9.96% 9.98% 10.13% 10.05% 9.96% 10.04%9.77% 9.80% 10.02% 10.17% 9.94% 9.89% 10.01% 10.09% 9.94% 10.06%9.78% 9.82% 10.21% 10.06% 9.99% 9.97% 9.76% 10.08% 9.87% 10.08%9.78% 9.77% 10.20% 10.14% 10.02% 10.02% 9.76% 10.09% 10.00% 10.09%9.71% 9.75% 10.23% 10.14% 10.02% 10.00% 9.84% 10.20% 9.90% 10.06%9.64% 9.65% 10.17% 10.12% 10.13% 9.95% 9.77% 10.25% 9.93% 9.90%9.56% 9.62% 10.22% 10.20% 10.14% 9.79% 9.73% 10.25% 10.03% 10.10%9.62% 9.60% 10.32% 10.11% 10.16% 9.82% 9.74% 10.22% 9.93% 10.17%9.62% 9.57% 10.33% 10.03% 10.19% 9.84% 9.69% 10.17% 9.84% 10.23%9.59% 9.49% 10.17% 10.03% 10.14% 9.82% 9.64% 10.13% 9.76% 10.19%9.62% 9.44% 9.98% 9.99% 10.11% 9.76% 9.66% 10.05% 9.96% 10.13%9.68% 9.48% 10.05% 9.95% 10.10% 9.87% 9.59% 10.10% 10.10% 10.02%9.17% 9.89% 9.63% 9.03% 10.48% 9.17% 8.66% 10.10% 10.64% 9.02%9.09% 9.88% 9.59% 9.00% 10.53% 9.23% 8.62% 10.07% 10.68% 8.94%9.09% 9.90% 9.46% 8.99% 10.47% 9.24% 8.54% 10.02% 10.54% 8.89%9.22% 9.83% 9.47% 8.94% 10.36% 9.29% 8.58% 10.06% 10.42% 8.85%9.35% 9.94% 9.43% 8.87% 10.40% 9.40% 8.58% 9.97% 10.40% 8.91%9.42% 9.93% 9.30% 8.83% 10.44% 9.39% 8.48% 9.96% 10.48% 9.08%9.23% 9.89% 9.23% 8.90% 10.42% 9.47% 8.50% 9.99% 10.45% 9.10%9.16% 9.86% 9.10% 8.97% 10.42% 9.43% 8.51% 9.88% 10.46% 9.11%

9.56% 9.61% 9.92% 9.64% 10.05% 9.71% 9.41% 10.30% 10.00% 9.37%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0



Sim 155 Sim 156 Sim 157 Sim 158 Sim 159 Sim 160 Sim 161 Sim 162 Sim 163 Sim 16410.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00%10.03% 10.08% 9.95% 9.96% 9.90% 10.11% 10.00% 9.94% 10.03% 10.06%

9.89% 10.01% 9.97% 9.95% 9.91% 10.24% 9.93% 9.95% 9.99% 10.04%9.98% 10.00% 10.00% 9.88% 9.95% 10.18% 9.95% 9.95% 9.89% 9.95%9.99% 10.06% 9.93% 9.74% 9.90% 10.12% 9.94% 9.92% 9.98% 9.88%9.94% 10.04% 9.78% 9.85% 9.80% 10.13% 10.00% 9.84% 10.03% 9.79%9.92% 9.96% 9.89% 9.67% 9.80% 10.08% 9.96% 9.68% 10.05% 9.77%9.88% 9.88% 9.94% 9.60% 9.78% 10.21% 9.93% 9.76% 10.08% 9.77%9.85% 9.95% 9.85% 9.64% 9.84% 10.15% 9.97% 9.74% 10.09% 9.83%9.87% 10.05% 9.83% 9.61% 9.88% 10.02% 9.97% 9.82% 10.12% 9.89%9.82% 10.15% 9.81% 9.72% 9.88% 10.04% 10.03% 9.81% 10.13% 9.97%9.82% 10.19% 9.87% 9.62% 9.98% 10.03% 10.04% 9.95% 10.04% 10.07%9.74% 10.21% 9.75% 9.40% 9.96% 9.91% 9.99% 9.95% 10.10% 10.08%9.70% 10.33% 9.79% 9.39% 10.08% 9.95% 9.89% 9.82% 10.15% 9.98%8.95% 9.65% 9.29% 8.07% 9.75% 9.05% 10.27% 9.42% 9.58% 10.00%8.90% 9.65% 9.24% 8.06% 9.87% 8.99% 10.17% 9.42% 9.61% 10.08%8.74% 9.72% 9.31% 8.10% 9.83% 8.99% 10.16% 9.42% 9.45% 10.08%8.67% 9.81% 9.34% 8.11% 9.75% 8.89% 10.20% 9.35% 9.38% 10.08%8.59% 9.89% 9.31% 8.15% 9.72% 8.87% 10.21% 9.42% 9.56% 10.11%8.59% 9.83% 9.41% 8.07% 9.82% 8.87% 10.20% 9.46% 9.51% 10.07%8.51% 9.88% 9.46% 8.17% 9.86% 8.83% 10.15% 9.54% 9.59% 10.05%8.42% 9.78% 9.48% 8.06% 9.93% 8.89% 10.16% 9.71% 9.63% 10.00%

9.42% 9.93% 9.73% 8.69% 9.79% 9.50% 10.25% 9.55% 10.04% 9.88%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.94% 9.86% 10.03% 9.98% 9.96% 10.01% 9.95% 10.03% 9.87% 10.05%9.99% 9.77% 10.04% 9.87% 9.98% 10.06% 9.96% 9.99% 9.92% 10.13%

10.09% 9.76% 10.08% 9.86% 10.14% 10.04% 9.92% 9.92% 9.83% 10.07%9.98% 9.83% 10.15% 9.93% 10.04% 10.15% 9.94% 9.88% 9.89% 10.19%9.98% 9.94% 10.12% 9.93% 9.99% 9.98% 9.91% 9.95% 9.91% 10.18%9.97% 9.94% 10.04% 10.01% 9.86% 10.00% 9.83% 9.93% 9.99% 10.17%

10.09% 9.91% 10.03% 10.10% 9.86% 9.99% 9.77% 9.84% 10.02% 10.14%10.02% 9.94% 10.06% 10.10% 9.90% 10.10% 9.79% 9.81% 10.02% 10.09%

9.95% 10.01% 10.04% 10.04% 10.00% 10.12% 9.80% 9.82% 10.07% 10.11%9.95% 10.00% 10.05% 10.05% 10.09% 10.15% 9.73% 9.80% 10.09% 10.20%

10.02% 9.97% 10.10% 10.05% 10.13% 10.17% 9.60% 9.83% 10.00% 10.27%9.92% 9.99% 10.06% 10.01% 10.11% 10.19% 9.62% 9.82% 10.01% 10.24%9.78% 9.90% 10.07% 9.97% 10.08% 10.16% 9.68% 9.87% 9.89% 10.29%8.75% 10.76% 9.39% 9.52% 9.65% 8.33% 10.42% 8.63% 10.10% 9.75%8.78% 10.75% 9.44% 9.53% 9.67% 8.21% 10.35% 8.60% 10.08% 9.83%8.72% 10.87% 9.45% 9.59% 9.66% 8.22% 10.47% 8.61% 10.14% 9.87%8.72% 10.83% 9.51% 9.52% 9.61% 8.23% 10.38% 8.58% 10.17% 9.88%8.70% 10.74% 9.43% 9.42% 9.52% 8.37% 10.24% 8.51% 10.19% 9.86%8.66% 10.67% 9.40% 9.45% 9.56% 8.47% 10.13% 8.45% 10.27% 9.71%8.78% 10.63% 9.39% 9.37% 9.48% 8.61% 10.08% 8.34% 10.19% 9.87%8.74% 10.64% 9.32% 9.37% 9.62% 8.40% 10.14% 8.33% 10.08% 9.94%

9.31% 10.61% 9.63% 9.70% 10.05% 9.61% 10.46% 9.64% 9.78% 10.12%

0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0.01 0 0 0 0 0 0 0 0



Sim 175 Sim 176 Sim 177 Sim 178 Sim 179 Sim 180 Sim 181 Sim 182 Sim 183 Sim 18410.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00%10.11% 10.02% 10.05% 9.84% 9.94% 10.06% 9.90% 9.90% 9.88% 10.01%10.10% 10.07% 10.11% 9.70% 9.96% 10.03% 9.99% 9.82% 9.95% 10.05%10.08% 10.20% 10.01% 9.80% 9.96% 10.03% 9.92% 9.69% 9.92% 10.02%10.06% 10.12% 10.02% 9.80% 9.91% 10.14% 10.09% 9.80% 9.81% 10.01%10.12% 10.23% 9.97% 9.81% 9.88% 10.18% 10.01% 9.85% 9.81% 9.95%10.02% 10.16% 10.09% 9.89% 9.84% 10.20% 10.03% 9.87% 9.79% 9.88%10.07% 10.16% 10.02% 9.87% 9.86% 10.24% 9.98% 9.81% 9.81% 9.94%10.05% 10.17% 10.02% 9.78% 9.91% 10.19% 9.98% 9.76% 9.95% 9.88%10.00% 10.06% 9.85% 9.77% 9.89% 10.13% 9.89% 9.68% 9.98% 10.00%10.03% 10.12% 9.89% 9.81% 9.80% 10.10% 9.85% 9.68% 9.92% 9.99%10.11% 10.10% 9.80% 9.77% 9.79% 10.15% 9.81% 9.75% 9.83% 9.99%10.07% 10.13% 9.73% 9.80% 9.91% 10.13% 9.73% 9.82% 9.74% 9.92%

9.92% 10.13% 9.66% 9.77% 9.94% 10.11% 9.73% 9.90% 9.73% 9.84%9.53% 10.19% 9.97% 9.88% 9.51% 9.46% 10.80% 9.41% 8.50% 10.40%9.54% 10.13% 10.03% 9.86% 9.44% 9.42% 10.83% 9.36% 8.43% 10.40%9.43% 10.05% 9.96% 9.82% 9.50% 9.47% 10.92% 9.43% 8.34% 10.34%9.39% 10.02% 9.99% 9.71% 9.52% 9.46% 10.94% 9.48% 8.34% 10.26%9.47% 9.94% 10.00% 9.78% 9.39% 9.37% 10.98% 9.45% 8.26% 10.17%9.51% 9.82% 9.92% 9.79% 9.46% 9.29% 10.91% 9.51% 8.36% 10.16%9.53% 9.77% 9.94% 9.84% 9.50% 9.39% 11.00% 9.43% 8.35% 10.18%9.60% 9.76% 9.96% 9.82% 9.49% 9.40% 11.01% 9.39% 8.36% 10.29%

9.86% 10.17% 9.79% 9.69% 9.46% 10.07% 10.21% 9.33% 9.30% 9.91%

0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00

0 0 0 0 0 0 0.01 0 0 0



Sim 185 Sim 186 Sim 187 Sim 188 Sim 189 Sim 190 Sim 191 Sim 192 Sim 193 Sim 19410.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00%10.02% 9.97% 9.98% 9.97% 9.92% 9.98% 9.98% 10.06% 9.96% 10.00%10.00% 9.93% 10.05% 10.01% 9.98% 9.87% 9.88% 10.09% 9.89% 10.08%

9.94% 9.95% 10.08% 10.07% 9.97% 9.95% 9.86% 10.06% 9.87% 10.09%9.90% 9.91% 10.10% 10.09% 9.94% 9.99% 9.81% 9.86% 9.83% 10.09%9.89% 10.07% 10.19% 9.99% 9.94% 10.06% 9.78% 9.74% 9.77% 10.14%9.85% 10.09% 10.18% 9.91% 10.02% 10.04% 9.88% 9.65% 9.75% 10.05%9.87% 10.27% 10.16% 10.06% 10.07% 9.97% 9.87% 9.62% 9.76% 9.91%9.83% 10.48% 10.04% 9.98% 10.12% 9.96% 9.86% 9.44% 9.86% 9.93%9.75% 10.44% 10.05% 9.86% 10.15% 9.91% 9.65% 9.50% 9.77% 9.89%9.80% 10.49% 9.96% 9.89% 10.18% 9.92% 9.44% 9.50% 9.69% 9.86%9.87% 10.49% 9.98% 9.91% 10.20% 9.92% 9.42% 9.42% 9.72% 9.83%9.90% 10.51% 10.01% 10.03% 10.25% 9.94% 9.39% 9.33% 9.71% 9.77%9.87% 10.47% 10.01% 10.00% 10.17% 10.12% 9.27% 9.29% 9.80% 9.80%9.62% 9.93% 9.37% 9.06% 9.06% 9.66% 8.94% 7.87% 9.69% 10.26%9.56% 9.96% 9.36% 9.12% 8.97% 9.71% 9.08% 7.84% 9.76% 10.21%9.49% 9.91% 9.36% 9.04% 9.05% 9.64% 9.04% 7.80% 9.68% 10.27%9.42% 9.88% 9.42% 8.93% 9.17% 9.70% 9.20% 7.85% 9.75% 10.26%9.47% 9.84% 9.44% 8.92% 9.05% 9.78% 9.30% 7.86% 9.85% 10.25%9.45% 9.87% 9.52% 9.01% 9.06% 9.79% 9.31% 7.86% 9.68% 10.24%9.47% 9.98% 9.51% 8.88% 9.08% 9.67% 9.24% 7.82% 9.61% 10.27%9.60% 10.01% 9.48% 8.81% 9.08% 9.59% 9.14% 7.83% 9.59% 10.39%

9.69% 10.26% 9.65% 9.78% 9.87% 9.86% 9.19% 8.76% 9.75% 10.09%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0



Sim 195 Sim 196 Sim 197 Sim 198 Sim 199 Sim 200 Sim 201 Sim 202 Sim 203 Sim 20410.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00%10.11% 10.10% 10.02% 10.02% 9.94% 10.02% 10.06% 9.97% 9.83% 9.96%10.15% 10.21% 9.94% 10.03% 9.93% 10.10% 10.07% 9.90% 9.87% 10.02%10.27% 10.06% 9.84% 9.96% 9.91% 10.12% 10.10% 9.86% 9.89% 9.99%10.26% 10.12% 9.90% 10.06% 9.83% 10.16% 10.24% 9.90% 9.81% 9.96%10.21% 10.04% 9.99% 9.99% 9.95% 10.14% 10.21% 9.84% 9.73% 9.96%10.21% 10.04% 9.85% 10.06% 9.90% 10.09% 10.15% 9.74% 9.71% 9.90%10.24% 9.99% 9.84% 10.22% 9.79% 10.13% 10.25% 9.82% 9.63% 9.85%10.14% 9.93% 9.75% 10.24% 9.82% 10.06% 10.11% 9.90% 9.65% 9.89%10.08% 10.08% 9.81% 10.17% 9.83% 10.22% 10.01% 9.92% 9.76% 9.88%10.01% 10.19% 9.72% 10.30% 9.90% 10.15% 10.00% 9.92% 9.68% 9.74%10.03% 10.05% 9.78% 10.39% 9.90% 10.24% 10.15% 9.95% 9.62% 9.70%

9.96% 10.16% 9.81% 10.37% 9.88% 10.11% 10.21% 9.87% 9.56% 9.68%9.95% 10.26% 9.82% 10.31% 9.89% 10.17% 10.22% 9.90% 9.59% 9.49%

10.00% 8.89% 9.67% 9.62% 9.93% 9.71% 9.18% 9.64% 9.66% 9.56%9.91% 8.94% 9.84% 9.56% 9.91% 9.74% 9.27% 9.65% 9.71% 9.61%9.85% 8.93% 9.79% 9.50% 9.81% 9.65% 9.28% 9.68% 9.59% 9.53%9.87% 8.94% 9.70% 9.44% 9.74% 9.66% 9.27% 9.72% 9.64% 9.53%9.89% 8.91% 9.52% 9.43% 9.74% 9.64% 9.19% 9.65% 9.60% 9.45%9.81% 8.96% 9.49% 9.51% 9.80% 9.69% 9.24% 9.53% 9.60% 9.47%9.70% 8.93% 9.44% 9.56% 9.77% 9.79% 9.19% 9.44% 9.52% 9.53%9.66% 9.10% 9.50% 9.66% 9.79% 9.86% 9.10% 9.47% 9.55% 9.45%

9.87% 9.64% 9.74% 10.09% 10.25% 9.90% 9.81% 9.95% 9.99% 9.59%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.87% 9.88% 10.00% 10.04% 9.82% 10.01% 9.99% 10.00% 10.02% 10.06%9.84% 9.90% 10.01% 9.83% 9.84% 10.08% 10.07% 9.95% 10.06% 9.95%9.85% 9.93% 10.04% 9.86% 9.92% 10.01% 10.08% 10.01% 10.07% 10.07%9.82% 10.02% 10.03% 9.83% 9.86% 9.93% 10.08% 9.93% 10.11% 10.08%9.86% 9.90% 9.95% 9.77% 9.90% 9.81% 9.94% 9.95% 10.05% 10.00%9.89% 9.83% 10.00% 9.88% 9.81% 9.80% 9.99% 9.88% 10.13% 10.03%9.83% 9.82% 9.92% 9.95% 9.77% 9.80% 9.95% 9.96% 10.16% 10.09%9.82% 9.73% 9.85% 9.94% 9.77% 9.70% 10.00% 9.90% 10.08% 10.07%9.78% 9.74% 9.82% 9.86% 9.79% 9.63% 9.96% 9.95% 10.13% 9.99%9.60% 9.83% 9.74% 9.90% 9.84% 9.58% 9.99% 9.93% 10.10% 9.99%9.66% 9.89% 9.78% 9.93% 10.00% 9.55% 10.00% 9.91% 10.11% 10.05%9.65% 9.85% 9.80% 10.00% 9.98% 9.60% 9.97% 9.82% 10.11% 10.15%9.71% 9.88% 9.73% 9.99% 10.02% 9.58% 9.95% 9.78% 10.14% 10.10%9.95% 8.83% 10.18% 9.24% 9.59% 9.62% 9.48% 10.69% 10.14% 10.25%

10.08% 8.78% 10.27% 9.26% 9.53% 9.72% 9.48% 10.67% 10.03% 10.33%10.08% 8.71% 10.37% 9.24% 9.55% 9.66% 9.48% 10.50% 9.99% 10.31%10.06% 8.84% 10.54% 9.15% 9.47% 9.65% 9.46% 10.48% 10.12% 10.35%10.03% 8.83% 10.54% 9.24% 9.54% 9.71% 9.43% 10.44% 10.06% 10.32%10.11% 8.77% 10.44% 9.21% 9.60% 9.70% 9.51% 10.40% 10.19% 10.22%10.07% 8.68% 10.51% 9.18% 9.55% 9.85% 9.59% 10.30% 10.16% 10.25%10.04% 8.62% 10.42% 9.25% 9.47% 9.89% 9.70% 10.30% 10.07% 10.19%

9.55% 9.30% 9.82% 9.77% 9.72% 9.78% 9.67% 10.15% 10.13% 10.24%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.99% 9.99% 9.89% 9.93% 9.94% 10.07% 9.94% 10.00% 10.07% 10.10%9.94% 9.88% 9.73% 9.86% 9.91% 9.93% 9.88% 9.99% 10.17% 10.08%9.98% 9.85% 9.71% 9.92% 9.87% 9.99% 9.84% 9.99% 10.01% 10.09%9.88% 9.77% 9.78% 9.92% 9.95% 9.90% 9.82% 10.10% 9.97% 10.08%9.90% 9.77% 9.73% 9.96% 9.86% 9.88% 9.78% 10.16% 10.04% 10.11%9.74% 9.77% 9.73% 9.85% 9.82% 9.88% 9.70% 10.10% 10.08% 10.11%9.60% 9.70% 9.69% 9.86% 9.82% 9.92% 9.55% 10.11% 10.17% 10.03%9.62% 9.66% 9.82% 9.84% 9.87% 10.02% 9.57% 10.07% 10.15% 10.03%9.59% 9.75% 9.79% 9.74% 9.90% 10.11% 9.55% 10.04% 10.12% 9.99%9.45% 9.72% 9.81% 9.65% 9.89% 10.01% 9.62% 10.10% 10.05% 10.08%9.57% 9.69% 9.83% 9.61% 9.83% 10.02% 9.57% 10.28% 10.03% 9.92%9.48% 9.79% 9.79% 9.53% 9.79% 10.00% 9.44% 10.17% 10.05% 9.99%9.58% 9.81% 9.87% 9.52% 9.96% 9.92% 9.55% 10.10% 9.98% 9.94%9.47% 9.52% 8.80% 8.79% 10.35% 9.20% 9.45% 9.24% 10.16% 8.50%9.46% 9.53% 8.64% 8.81% 10.37% 9.15% 9.47% 9.38% 10.02% 8.45%9.48% 9.48% 8.54% 8.80% 10.39% 9.14% 9.43% 9.34% 10.03% 8.36%9.34% 9.52% 8.55% 8.78% 10.45% 9.08% 9.39% 9.41% 10.04% 8.45%9.30% 9.43% 8.52% 8.90% 10.48% 9.07% 9.46% 9.38% 10.09% 8.48%9.27% 9.49% 8.59% 8.92% 10.43% 9.06% 9.39% 9.25% 10.17% 8.43%9.30% 9.41% 8.60% 8.88% 10.46% 9.09% 9.35% 9.33% 10.13% 8.39%9.25% 9.37% 8.57% 8.98% 10.45% 9.08% 9.34% 9.32% 9.96% 8.41%

9.54% 9.52% 9.27% 9.29% 9.99% 9.85% 9.50% 9.87% 9.97% 9.29%

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0 0 0 0 0 0 0 0 0 0




9.99% 10.09% 9.90% 10.06% 10.01% 10.02% 9.84% 9.97% 10.02% 9.99%10.04% 10.02% 10.02% 10.08% 9.99% 9.98% 9.90% 9.84% 9.97% 9.94%10.08% 10.10% 9.90% 10.08% 10.07% 10.04% 9.99% 9.74% 9.87% 9.97%

9.96% 10.07% 9.82% 10.13% 10.15% 10.08% 9.94% 9.79% 9.88% 9.98%9.89% 10.00% 9.68% 10.16% 10.20% 10.00% 10.05% 9.75% 9.85% 9.87%9.85% 9.86% 9.64% 10.12% 10.14% 10.03% 10.02% 9.80% 9.78% 9.94%9.92% 9.93% 9.73% 10.19% 10.12% 10.18% 9.93% 9.78% 9.70% 9.95%9.94% 10.01% 9.68% 10.22% 10.02% 10.35% 9.87% 9.81% 9.60% 9.95%9.91% 10.03% 9.71% 10.23% 10.03% 10.48% 9.87% 9.80% 9.52% 9.85%9.89% 10.03% 9.75% 10.28% 10.03% 10.43% 9.88% 9.73% 9.45% 9.88%9.83% 9.97% 9.76% 10.41% 10.02% 10.34% 9.80% 9.71% 9.47% 9.90%9.81% 10.08% 9.77% 10.45% 9.95% 10.30% 9.92% 9.61% 9.46% 9.88%9.77% 10.11% 9.70% 10.46% 9.87% 10.42% 9.84% 9.60% 9.42% 9.81%8.36% 10.00% 9.93% 8.67% 9.76% 10.78% 9.92% 8.12% 10.19% 10.20%8.42% 9.97% 9.98% 8.72% 9.89% 10.74% 10.13% 8.08% 10.09% 10.19%8.48% 9.99% 9.98% 8.76% 9.90% 10.66% 10.19% 8.09% 10.06% 10.20%8.36% 9.84% 9.91% 8.82% 9.88% 10.57% 10.13% 8.11% 10.06% 10.26%8.35% 9.75% 9.87% 8.83% 9.89% 10.64% 10.23% 8.08% 10.02% 10.34%8.32% 9.62% 9.86% 8.76% 9.91% 10.77% 10.17% 8.15% 10.03% 10.24%8.25% 9.62% 9.75% 8.88% 9.88% 10.75% 10.24% 8.17% 10.09% 10.29%8.29% 9.65% 9.81% 8.90% 9.85% 10.73% 10.06% 8.06% 10.05% 10.18%

9.13% 9.98% 9.81% 9.24% 9.93% 10.69% 9.83% 9.08% 9.54% 10.13%

0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00

0 0 0 0 0 0.01 0 0 0 0



Sim 245 Sim 246 Sim 247 Sim 248 Sim 249 Sim 250 Sim 25110.00% 10.00% 10.00% 10.00% 10.00% 10.00% 10.00%

9.99% 9.98% 10.02% 10.05% 10.02% 9.95% 10.16%10.06% 10.00% 9.95% 10.02% 9.98% 10.08% 10.15%10.04% 9.97% 10.04% 10.05% 9.94% 10.12% 10.20%

9.94% 9.92% 10.10% 10.06% 9.90% 10.11% 10.18%9.87% 9.81% 10.21% 10.14% 9.84% 10.08% 10.10%9.74% 9.74% 10.20% 10.21% 9.81% 9.96% 10.11%9.82% 9.78% 10.19% 10.20% 9.77% 9.95% 10.15%9.89% 9.71% 10.28% 10.11% 9.70% 9.98% 10.24%9.86% 9.77% 10.26% 10.03% 9.66% 10.05% 10.11%9.77% 9.76% 10.30% 10.05% 9.64% 9.94% 10.03%9.76% 9.80% 10.41% 10.07% 9.66% 9.98% 10.02%9.72% 9.82% 10.37% 10.03% 9.75% 9.95% 10.09%9.80% 9.69% 10.38% 10.03% 9.85% 9.98% 10.06%9.37% 9.58% 10.61% 9.74% 9.35% 9.41% 10.29%9.37% 9.59% 10.74% 9.73% 9.39% 9.48% 10.25%9.34% 9.53% 10.63% 9.70% 9.34% 9.47% 10.23%9.31% 9.54% 10.53% 9.65% 9.28% 9.53% 10.10%9.20% 9.51% 10.59% 9.65% 9.21% 9.55% 10.03%9.20% 9.58% 10.65% 9.59% 9.12% 9.60% 10.06%9.27% 9.44% 10.58% 9.53% 9.10% 9.70% 10.10%9.32% 9.57% 10.64% 9.41% 9.10% 9.60% 10.05%

9.84% 9.50% 10.18% 10.03% 9.76% 9.83% 9.96%

0.00 0.00 0.01 0.00 0.00 0.00 0.00

0 0 0.01 0 0 0 0



© Paul Wilmott 2001Demonstration of Cholesky factorization. Inputs in grey boxes

Input symmetric (positive definite) correlation matrix

0.45 0.03 0.13 0.06 0.090.03 0.34 0 -0.09 0.120.13 0 0.28 0 -0.130.06 -0.09 0 0.41 0.030.09 0.12 -0.13 0.03 0.32

Matrix M #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

Product of M #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!and M^T #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! This should be the s#VALUE! #VALUE! #VALUE! #VALUE! #VALUE! as the input matrix#VALUE! #VALUE! #VALUE! #VALUE! #VALUE!



me



© Paul Wilmott 2001Numerical integration

Payoff 2 e.g. 2 = 2nd highest out of 5Calculation type

Interest rate 0.08Expiry 1 See VB code

No. iterations 10000Value 106.6429

Asset Price Vol. Div. YldA 100 0.2 0.02B 100 0.2 0.02C 100 0.2 0.02D 100 0.2 0.02E 100 0.2 0.02

Correlations1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

þÿMonte Ca rlo



Monte Carlo 1Quasi Monte Carlo



lo



50 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!51 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!52 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!53 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!54 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!55 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!56 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!57 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!58 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!59 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!60 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!61 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!62 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!63 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!64 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!65 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!66 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

67 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!68 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!69 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!70 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!71 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!72 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!73 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!74 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!75 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!76 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!77 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!78 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

79 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!80 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!81 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!82 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!83 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!84 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!85 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!86 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!87 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!88 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!89 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!90 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!91 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!92 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!93 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!94 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!95 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!96 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!97 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!98 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!99 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

100 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!101 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!102 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!103 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!





133 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!134 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!135 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!136 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!137 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!138 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!139 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!140 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!141 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!142 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!143 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!144 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!145 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!146 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!147 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!148 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!149 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!150 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!151 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!152 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!153 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!154 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!155 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!156 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!157 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!





997 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!998 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!999 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!




0.8 0.9 1

plot of?

monte carlo simulation (wilmott varie)

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